Finite Matrix Groups 1
399 views Jul 6, 2023
We advocate a more explicit concrete approach to finite group theory, where we begin by defining the objects in terms of matrix algebra, and look towards rich geometry that arises as a result. In this introductory lecture, we overview the basic definitions, some elementary results, and concentrate on some simple but important families of examples, including the Klein 4-group, cyclic and dihedral groups, the symmetric and alternating groups and the Weyl groups coming from ADE graphs via the Mutation and Numbers games. However you will be encouraged to think about these familiar objects in a somewhat novel, and completely specifiable, way. While this approach has significant overlap with the traditional classical approach to finite group theory and the associated representation theory, there are also key differences. One is that quite a bit of the rich geometric and combinatorial structures associated to Weyl groups can be extended to this more general situation. Another is that the linear algebra supports notions usually more associated to Lie group theory and harmonic analysis to come to the fore. This will be the first of a series of exploratory lectures on this topic, meant for Members of the Wild Egg maths channel. Patreon supporters will also have access to this content. This is one of several Playlists exploring Research Level Mathematics meant for amateur mathematicians and prospective pure mathematicians, and even for practicing academics, as we will be looking at many things in a new, much more computational and explicit light. Not for us abstract invocation of "mathematical objects" which cannot be specified completely and concretely on a computer!
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Hello everyone I'm Norman thanks for joining me on this new adventure in exploring Research Level
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Mathematics today we're starting something really exciting we're going to be looking at Finite Group
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Theory very classical beautiful and much studied subject in a new way by connecting more strongly
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with linear algebra and some attendant geometry so I'm proposing to think about
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finite groups as matrices okay so this connects naturally with an aspect of
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finite group theory called Representation Theory so in some sense this is a natural
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and important part of traditional group theory but my orientation is that by starting with
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the matrices as the central objects we find all kinds of new ways of thinking
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about things and we find new structures that are not so easily visible in the usual story
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now my motivations for starting this course are twofold one is the Dynamics on Graphs
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Series where I've been introducing the ADE graphs and mutation and numbers games and if you've been
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watching that which I hope you have you'll know that there are these mutation matrices which play a central role and Associated to that there are Cartan matrices there's the Tits
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quadratic form both on populations and weights and there's all kinds of interesting connections
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with mathematical physics so motivated by that and and the development of that I've laid out I'm seeing that some of those notions that are appearing there are actually
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easily generalizable so I want to take some of those ideas and try to generalize them much
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more broadly so they apply to finite groups more generally okay so that's one of my motivations
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the other motivation is perhaps in a quite a different direction coming from my background as a harmonic analyst or a Lie group representation theorist so I used to study Lie groups a lot I'm
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still very interested in those things and while Lie algebras are finite in some sense finally
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specifiable because it's all about linear spaces Lie groups themselves are much more complicated
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you need to have some background in differentiable manifolds and vector fields and exponential maps
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and so on and to be honest the specification of Lie groups is a highly problematic
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subject so going back to this distinction between descriptions and specifications
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um there's lots of ways of describing Lie groups you know it's a group of isometries of such and such structure or such but to connect it directly with the actual official definition to specify a
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lead group that's much more challenging and so I'm I think I'm not the only uh sort of person who has
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you know come to this realization that a really nice way of thinking about Lie groups is to think
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of them as matrices right from the the get-go okay that's a very powerful point of view which
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is not the mainstream but it's also uh you know has been explored and so I'm going to be taking
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that orientation and applying it to the finite group situation okay so uh you can read what
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I've said here so I'm going to start today with this um this introductory talk so I'm just going
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to lay out just a few definitions so we're all on the same page in terms of some basic terminology
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and then I want to look at some main examples of these finite matrix groups so these examples are
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going to be very familiar to you if you've taken a classical course in finite group theory but you're
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going to see them in a novel way and they're going to be pretty interesting for us okay so
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the the examples are first and foremost we want to have a wide range of examples so that we can have
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things to investigate okay so in terms of basic of terminology by a number I mean a rational number
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so the field of rational numbers is our sort of defaults field but you know down the road we
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also want to think about finite fields and maybe complex rational numbers maybe other fields as
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well so the question is to how much of what I'm talking about extends to other fields is is an
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interesting one and I think that's also going to be interesting for you to explore okay but let's stick at least now for the with the rational number case so the main objects are going to
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be from linear algebra a matrix okay but I'm going to use the term an n Matrix okay so for a
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natural number n greater than or equal to one an n matrix is an n by n array of rational numbers
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the spaces of n matrices forms an algebra mat and let's call it over the rational numbers with
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identity Matrix I_n that's the n by n identity Matrix with ones on the diagonal zeros elsewhere
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and I remind you that an n matrix a is invertible precisely when it's determinant is non-zero
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and sometimes we'll also talk about vectors so we'll talk about row n vectors or column
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n vectors in the obvious sense okay so that's just some brief linear algebra we
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are going to have to touch base with Matrix Theory much more extensively as we carry on
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because we want to frame everything ultimately in terms of this linear algebra setting okay so
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what's a finite Matrix Group well let's say you have a natural number n greater than or equal
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to one a finite n Matrix Group okay so that's the terminology a finite and Matrix Group G is
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a finite set of n matrices which is closed under multiplication and taking inverses so you take
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any two of these n matrices you multiply them you get another one which is still in the set
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and you take the inverse of any one of them you get another one which is still in the set
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in particular this set G contains the n by n identity Matrix I sub n
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the product of matrices which will often denote by this this x here it's automatically associative
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because matrix multiplication is so that we don't have to say that uh separately
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in terms of some notation we'll say that g is of dimension n because the matrices themselves are
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n by n and we'll write n equals dimension G and if G has k elements the number of elements then
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we'll say it's of order k and we'll write k as order of G or just absolute value of G
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and if the dimension n is understood or may be variable we'll just refer to G as being a finite
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Matrix Group and we'll introduce a shorthand for that f.m.g. standing for finite Matrix Group
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okay a few uh basic things a commutative finite Matrix Group well that's just one g times h is h
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times h for any elements of G and H and in such find it and Matrix Group we have left and right
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also right cancellation if g times h equals g times k then h equals h and the argument
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is familiar if you have this relation then you multiply on both sides by the inverse of G which
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is the existence by our definition and then by associativity you rearrange this to get h equals h
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and symmetrically also we have right cancellation we have h times g equals k times g then h equals k
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Another pretty simple consequence of things is that in a finite n Matrix Group each element g is a matrix of finite order meaning that there's some natural numbers
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say r such that if you raise g to the r-th power you're going to get the identity Matrix
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and well the reason is so basically you take this element g okay and if the size of G is
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k might as well assume that k is bigger than 1 If K is equal to one we're just talking about the identity Matrix which obviously has order one already so otherwise you just take this element
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g and you look at the various powers of it g g squared g cubed up to g to the k plus one okay
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and these are k plus 1 elements in this set which has k elements so by the pigeonhole principle
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there must be some repetition too these guys have to be the same so that means we can write g to the
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l equals g to the , for some numbers l and m we might as well assume that l is less than m and
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then by multiplying by gl to the minus one on both sides of this we get that g to the m minus
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l is equal to the identity Matrix so we've found a power of g which is the identity matrix so this
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makes it clear that the matrices that are involved in finite Matrix groups are going to be rather
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special they're matrices of finite order so we'll have to learn a little bit about those things
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another kind of standard thing to do when you've defined in algebraic structures you define some subgroups so if H is a subset of a finite n matrix group G
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and if H is also closed under multiplication and taking inverses then H is also a finite
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n matrix group and we'll say that H is a sub group of G and also that G is a super group of H
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this letter terminology is not at all as common as a subgroup but it makes things a little bit more
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symmetrical so H is a subgroup of G equivalent to G is a super group of H and we'll write less than
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or equal to for this relation so h less than or equal to g means that H is a subgroup of G
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and or equivalently G is a super group of H and this allows for the possibility of equality and
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this is clearly transitive if H is a subgroup of G and G is a subgroup of L then H is a subgroup of L
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Now in terms of determinants since the basic rule for a determinant of a matrix is that
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the determinant of a product is the product of the determinants it means that if you take the determinant of a power of some Matrix say x to the n that's the same as the determinant of x to the n
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and if we're talking about some Matrix X which has finite order so that some power of it is
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the identity Matrix which has determinant one it means that the determinants of such
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things have to be strongly constrained right because the determinant has to have the property that if you raise it to some power you're going to get one
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now in the rational numbers there's only two such numbers that have that property namely one
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or minus one so so any element of a finite Matrix Group must have determined either one or a minus
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one and then if we look at the set of matrices with determinant equal to one so let's say we
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have a F.M.G. called G and we consider the set of matrices with determinant equal to one that's
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forming a subgroup let's say G_0 of G a subgroup because if you multiply two of them determined
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one the product will also be of determinant one okay so we'll call this the unary subgroup of G
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and okay this is in fact normal um we don't actually know what normal is
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um maybe I'll I'll just remove that for that we don't know what normal means yet um if K is uh any element of G which is not in this unary subgroup then the map g to g times
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K okay is a bijection between G zero and its complement because if g has determinant one
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and k has determinant minus one then g times k will also have determinant minus one and you can
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go backwards by multiplying by k inverse which will also have determined minus one so that's
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showing that the unary subgroup of the F.M.G. G is going to be exactly half the size of the big group
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okay so the size of G or the order of G is two times the order of G_0 now it's that's that's if
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there's an element in G which is not uh in G_0 if however G equals G_0 so in other words if all the
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elements of our F.M.G. are having determinant one we'll say that G is a it's a unary F.M.G.
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and uh if G and H are finite n Matrix groups
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then their intersection is also a finite n matrix Group okay so that's uh
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sort of obvious you have one set of matrices and another set of matrices but they're all in the
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world of n matrices then the intersection is going to have the property it's going to be closed under
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multiplication and taking inverses because each of the two individual sets is closed under those two things and so this G intersect H is going to be a subgroup of both G and H
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A much more subtle issue here concerns what happens when you you have these two finite n
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matrix groups G and H can we find some larger finite n matrix group K which contains both of
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them? Are they subgroups of a common finite n matrix group K okay so I don't know the
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answer to that question or how to approach it in general so conditions on H and G that guarantee the existence of such a sort of a common super group seems like a major challenge this is the
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kind of question that doesn't really make sense in ordinary finite group Theory okay because in
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finite groups if you have two finite groups they're not connected in any obvious way but here we're assuming we have two finite n matrix groups so they're all sitting in the world of n
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by n matrices so we can ask this question and if there is such a k okay then there will be
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a minimal such k and uh it's interesting to to ask you know whether this kind of thing happens
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um so some some related questions in this direction so given a finite n Matrix Group G can we always find a proper supergroup K in other words an F.M.G. also an n matrix
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group K such that G is a subgroup of K but G is not equal to K so can you always extend
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are there finite Matrix groups G which have no proper super groups that can't be extended so
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maximal ones and um are there F.M.G.s G which have a non-zero but finite number of proper
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super groups if so can we characterize such and in case G does have this sort of finite
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super group property then the collection of supergroups will be closed under into sections
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and so forms a potentially interesting post set ordered by inclusion and it's natural to extend
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this also to the post set of subgroups of G so giving us a post that contains strong information
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can we come up with some interesting examples of such so these questions already suggest to us that
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this theory is going to be a little bit different from usual finite group Theory and I suspect
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it's going to be really fertile ground for investigation by amateur mathematicians.
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So at the heart of any mathematical theory there are examples and it's really important
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to have a wide range of examples so that we can really freely explore
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the various aspects of whatever Theory we have in question right so let me give you some advice if
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you don't have a lot of examples in some proposed Theory steer clear from it okay it's not perhaps a
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worthwhile place for your investigations look for a mathematics where there's rich examples so with
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finite matrix groups there's lots of rich examples let's have a look at some of them starting with a very simple one so the simplest finite matrix groups have just one element the identity matrix
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okay so we could define for a natural number n greater than or equal to one the identity n group
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okay T_n let's say for like anything else T_n which just consists of the single identity matrix
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I_n okay there's only one product here I sub n times I sub n equals I sub n and only one inverse
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okay so it's clearly a group a little bit more complicated just a little bit more complicated
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is what we might call the parity n group where you consider the identity matrix I_n and it's negative
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and I often like to denote negative with a bar over things so I_n bar that's the same as minus
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I_ n and here is the multiplication table of this F.M.G. which is a useful construct
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okay where we just record the various possible products with the convention that if we're in a
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row indexed by a group element a and the column indexed by group element b then what should be
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here is the product of this one times this one so this element here is the product of is I_n
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times this I_n bar okay and here's the uh the obvious multiplication table for it and clearly
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just in the top left corner this you see the multiplication table for the earlier identity
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and group T_n so T_N is a subgroup of P_n and um in fact T_n equals P_n 0 is that really correct
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um uh no that's not actually correct is it because it's correct some of the time
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if n is even okay if n is even then this negative I sub n will still have determinant equal to one
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so um foreign it will not um there will not be the case so let me just modify that and say clearly
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T_N is a subgroup of P_n so the next example is the Klein 4 matrix group with these four
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four by four matrices the identity this one here we might call them a this one we might call it B
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this one might call C these are all permutation matrices in the sense that there's a single one in each row and column if we act on the right say on four vectors on row four vectors
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then this one keeps everything where it is this one here interchanges the first
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and second elements along with the third and fourth elements while this one interchanges the first and third elements along with the second and fourth elements while this one changes the
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first and fourth elements and the second and third elements and it's easy to see that the
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multiplication table is like this okay we see this is a little commutative group with four elements
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For our next example let's introduce first of all the notion of a cyclic n matrix for n a
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natural number greater than or equal to one and let's say that k is some number between 0 and n
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minus 1. So the cyclic n matrix C sub n k it's an n by n matrix okay whose i j - th entry is either
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going to be one or a zero and it's going to be one precisely when i minus j is congruent to k mod n
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so um when say n is three so here's the case when n equals three when k is zero
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this condition is i minus j is congruent to zero that means i equals j mod n
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that's giving us ones along this diagonal when k is equal to one say that i is equal to j plus one
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so the I is here this is like the two one entry right the I is 2 the j is one this is
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the three two entry the three is one more than two and this is the one three entry the one is
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one more than three which is the same as zero mod three and here is c 3 comma two so these
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matrices are constant on diagonals; diagonals sort of parallel to the main diagonal and have a
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sort of a diagonal of ones and someplace and then and then uh also something else um another one so
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here's also some examples um with C4 so now we can define the cyclic n matrix group okay where you can define the cyclic group in this context in this
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Matrix context so for a natural number n greater than or equal to one the cyclic n Matrix Group C_n that's the name of the group okay it's going to consist of the set of such
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matrices that we just talked about the cyclic n matrices C _n, k where k runs from zero up to n minus one
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okay so when n equals one there's only one such matrix the identity matrix the one by
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one identity matrix when n is two we have this is C sub 2 comma zero this is C sub 2 comma one
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C sub three well we're just getting these three matrices that we've already seen and C sub 4 we
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get these four matrices so we start with this diagonal of ones and then we sort of move that
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diagonal down one okay so now the diagonal is there but also one of them goes up to the top
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there okay and then we move it down again so we get this one here and we get another corresponding diagonal there and then here and here okay so these are the cyclic n matrix groups
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and it's an easy exercise to see that the product of any two of these is an is another one in fact c n comma k times c n comma l equals c n comma m, where m is just the sum k plus l mod n
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so in fact yes this is a commutative group and in fact it's a cyclic group meaning that each
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element is a power of some single element that single element being called a generator of the
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cyclic F.M.G. in this case an example of such a generating element it's not unique but an
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example is c n comma one like this one here is a generator of this group because if you take this
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thing here and you raise it to the second power you're going to get this thing you raise it to the third power you get this state you raise it to the fourth power you're going to get um the identity
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And in fact any cyclic F.M.G. whenever you have a group or an n matrix group in this context of
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uh which which has a cyclic property that there's a particular element so that all the others are
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powers of that particular element then that's necessarily commutative because powers of a
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single matrix always commute gives a matrix X and you look at its various powers x squared x cubed
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etc they always commute so this is a commutative group in general now you might think you know
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if you're sort of a classical a group theory a student you've learned about the cyclic groups
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um if we view them as multiplicative groups what would be your sort of incarnation of a
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cyclic group chances are you're going to say oh um we can realize these things as the nth roots of of
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unity in the complex plane where we take the unit circle in the complex numbers and we look at the
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nth roots of unity so we're subdividing the circle you're creating a regular polygon with n sides but
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other lectures of mine you'll know that creating that regular polygon is a highly problematic thing
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okay you can say consider the regular 11-gon but how do you construct that regular 11-gon
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so actually what you really need is in order to specify the X and Y coordinates of the vertices
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you need things like you know cosine of 2 pi over 11 or something you need to create a an infinite
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process in order to get the X and Y coordinates of these these vertices so that description of
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what a cyclic group is is in fact very problematic it's implicitly invoking some infinite processes
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from analysis that's what you're really doing so this approach here where we embed the cyclic
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groups as matrices yeah okay maybe they're a little bit big but it's completely algebraic
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and of course it connects with the the usual sort of interpretation of a cyclic group is sort of cyclically rotating so for example in the three uh n equals three case we take one
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of these matrices here like this one here and act on a row vector well it's going to rotate the vector okay but if you act on a column vector it's also going to rotate the elements
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but in sort of an opposite way so whether it's a kind of a clockwise or anti-clockwise rotation uh you know depends on whether you're acting on the row vectors on the left or column
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vectors on the right and similarly here's the other one it does sort of the opposite so
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cumulatively this cyclic group C sub 3 acts on row vectors as group of rotations it acts on column
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vectors also as a group of rotations or cyclical rotations but it does so in kind of different ways
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okay so you know cumulatively the effects are the same and this is an interesting point that
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that we'll be seeing over and over again here in this subject that when you have a Matrix Group
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there are really these two actions okay there's the action on the row vectors and there's the action on the column vectors and neither of these is intrinsically preferable it's exactly the same
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story that we saw in the Dynamics on Graph Series right where we have mutations on one side and
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dual mutations or firings on the other side and it's the interaction between these two
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actions if you like that generates so much of the interest of the story okay so you can see
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that we're we're looking at things a little bit differently from classical finite group theory
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course okay excellent so we have the cyclic groups what about the dihedral groups okay
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uh the dihedral n matrix groups D sub n these are super groups of the C n matrix groups okay and
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they're obtained by adding another n matrix which one well a rather special one also which we might
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call the cross diagonal identity and matrix L sub n okay and that's matrix like this that just has
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sort of ones on this sort of cross diagonal not going this way but going this way okay so
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what we're doing is we're adding this thing to the cyclic groups that we or we defined and then also
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of course necessarily adding products obtained by taking products of this thing here with the
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cyclic group elements and it turns out that that gives a bigger group which is exactly twice the
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size of the original cyclic group so for example if we have uh you know D3 so there's the original
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cyclic group C sub 3 we've added this one here and that's the product of this with this if you take
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the product of this with this you get this one we take the product this one this you get this one
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and here is D4 where you have the original C4 there and you're adding this cross diagonal
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guy L sub 4 and then here are various possible products of this times this this times this this
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times this giving you three more matrices now a question would be well how do we know this
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is closed so multiplicative closure of these dihedral groups follows from this relation that
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when you when you multiply one of the C and K matrices by the L sub n on the right you
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also get a product of L sub n on the left times another C matrix but c n comma n minus k so the
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the reflected or the symmetrical one on the other side so it's it's the presence of this
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relation and together with the fact that L_n squared itself has order two that allows you
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to check that these dihedral groups actually are closed under multiplication and inverses so that
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we really have honest F.M.G.s that's a really good exercise for you so please check this claim
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okay there's a special case that's pretty interesting this d sub 4
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um take cases particularly interesting that's um as it is a super group of the Klein 4 matrix group
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and connects in an important way with the algebra of two by two matrices and I have some lectures
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on Dihedrons so here's another manifestation of it in the sense that the multiplication table of the
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following agrees up to relabeling with the that of the D_4 group that I just defined so later on
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we'll talk about isomorphisms and homomorphisms and we'll be able to say something you know in in that language but at this stage here you should just think about this here's another
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okay which is is different from from this one these are four by four matrices this one here
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is a bunch of two by two matrices but nevertheless they have essentially the same algebraic structure
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and then if you wrote down multiplication tables for them you'd see that they're corresponding
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the multiplications are really the same up to re-labeling okay so what do we have here well we
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have uh matrices e, i, j, k these will be somewhat familiar if you've watched my dihedron lectures
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and over here these the next four are just the negations of these ones here so with some bars in
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them so we might call them e bar i bar j bar k bar and uh so these uh this one here maybe we call it
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D sub 4 2 or something it's actually isomorphic uh to the previous D sub 4 but not equal
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and and here's when you look at it this way you can sort of see the the role of the the dihedral
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group because when you act say on row vectors X and Y then you can see that the action of these
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guys is like a traditional dihedral group acting in the plane as rotation symmetries rotation and
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reflection symmetries of of a square in the plane and as a little exercise you might actually try
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to figure out where is the Klein matrix 4 group sitting inside here as a subgroup of this thing
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okay now we come to what's pretty well clearly the most important general family of examples
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of F.M.G.s playing a huge role in the entire subject and that's the symmetric groups but in
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our case the symmetric and matrix groups okay and S_n so what's that consist of well that
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consists of the permutation matrices which are of size n by n and a permutation matrix is an n by n
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matrix having exactly a single one in each row and column otherwise the entries are zero there
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are obviously n factorial such matrices and for the low dimensional cases we we can write them down easily enough so there's two of them in S2 there's one in this one and here are the six
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permutation matrices a three by three case now let me just reiterate okay that I've already
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talked about this at some length in the Dynamics on Graph course but we should think about these
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matrices as not exactly groups of permutations of a particular set okay so a matrix like this
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can act on row vectors acts on the right on row vectors it acts on the left on column vectors
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okay so yes we can interpret these things as permutations of a set maybe the set of row vectors
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or permutations so the set of column vectors but in fact you could also act with these matrices
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on other objects like other size matrices so it's really not the action as as permutations of a set
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that's primary for us here with this point of view what's primary is just thinking about these things
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as matrices and that's something that I've already emphasized in the Dynamics on Graphs of course
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and closely related to these are the alternating n matrices the alternating n matrix group A sub
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n okay which is the unary subgroup of S_n that is we look at the subset of these things which
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have determinant equal to one okay so for low dimensional cases what we have uh the A_1 is
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just one the A_2 is just a single element the A_3 just consists of these first three guys okay so
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the determinant one I think they correspond to the the rotations and the usual story the other
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ones sort of correspond to reflections and in A_4 instead of having 24 elements we have 12 elements
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and here they are and these are the matrices in the alternating group a sub 4. and finally
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our last example is coming from the Dynamics on Graphs course which is really a motivation for
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this whole story where you may remember when we were talking about the ADE graph A2 just
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two vertices joined by the single edge the mutation matrices s_1 and s_2 generate the
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Weyl Group which in this case is this group G with six elements containing also the these
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other two by two matrices so this is the group of the Weyl group in this case formed from the
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mutation game on on the graph A2 and of course as a group this turns out to be the same as S3
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okay but in general what we can do is for any ADE graph okay we can create a Weyl group in a similar
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way because we know that for the ADE graphs the mutation matrices generate a finite group okay we
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have these mutation matrices one for each vertex of the graph okay and then the group they generate
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is the Weyl group and that's a finite group if we start with a graph X which is an ADE graph so
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there's lots of interesting ADE graphs and and so that generates all kinds of interesting
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F.M.G.s for us the Weyl the classical Weyl groups coming from the ADE graphs
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so it's actually this example here that we're particularly interested in sort of
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generalizing and we want to see to what extent do the remarkable things that are happening in
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this story over here that I'm describing in the Dynamics of the Graphs course to what extent are
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there analogs of that for these more general F.M.G.s so these examples that we have here
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today is a good starting base of good grounding for investigating these finite matrix groups.