A Commutator

<ol style="margin: 0px; padding: 0px 0px 0px 2.25em; border: 0px; font-variant-numeric: inherit; font-variant-east-asian: inherit; font-stretch: inherit; line-height: inherit; font-family: -apple-system, BlinkMacSystemFont, "Segoe UI", Roboto, Lato, Helvetica, Arial, sans-serif; vertical-align: baseline; background-image: none; background-position: initial; background-size: initial; background-repeat: initial; background-attachment: initial; background-origin: initial; background-clip: initial; color: rgb(34, 34, 34);"><li style="margin: 0px 0px 10px; padding: 0px; border: 0px; font-style: inherit; font-variant: inherit; font-weight: inherit; font-stretch: inherit; line-height: inherit; font-family: inherit; vertical-align: baseline; background: none;">An <a href="https://en.m.wiktionary.org/wiki/electrical" title="electrical" style="margin: 0px; padding: 0px; border: 0px; font-style: inherit; font-variant: inherit; font-weight: inherit; font-stretch: inherit; line-height: inherit; font-family: inherit; vertical-align: baseline; background: none; color: rgb(107, 75, 161); hyphens: none;">electrical</a> <a href="https://en.m.wiktionary.org/wiki/switch" title="switch" style="margin: 0px; padding: 0px; border: 0px; font-style: inherit; font-variant: inherit; font-weight: inherit; font-stretch: inherit; line-height: inherit; font-family: inherit; vertical-align: baseline; background: none; color: rgb(107, 75, 161); hyphens: none;">switch</a>, in a <a href="https://en.m.wiktionary.org/wiki/generator" title="generator" style="margin: 0px; padding: 0px; border: 0px; font-style: inherit; font-variant: inherit; font-weight: inherit; font-stretch: inherit; line-height: inherit; font-family: inherit; vertical-align: baseline; background: none; color: rgb(107, 75, 161); hyphens: none;">generator</a> or <a href="https://en.m.wiktionary.org/wiki/motor" title="motor" style="margin: 0px; padding: 0px; border: 0px; font-style: inherit; font-variant: inherit; font-weight: inherit; font-stretch: inherit; line-height: inherit; font-family: inherit; vertical-align: baseline; background: none; color: rgb(107, 75, 161); hyphens: none;">motor</a>, that <a href="https://en.m.wiktionary.org/wiki/periodically" title="periodically" style="margin: 0px; padding: 0px; border: 0px; font-style: inherit; font-variant: inherit; font-weight: inherit; font-stretch: inherit; line-height: inherit; font-family: inherit; vertical-align: baseline; background: none; color: rgb(107, 75, 161); hyphens: none;">periodically</a> <a href="https://en.m.wiktionary.org/wiki/reverse" title="reverse" style="margin: 0px; padding: 0px; border: 0px; font-style: inherit; font-variant: inherit; font-weight: inherit; font-stretch: inherit; line-height: inherit; font-family: inherit; vertical-align: baseline; background: none; color: rgb(107, 75, 161); hyphens: none;">reverses</a> the <a href="https://en.m.wiktionary.org/wiki/direction" title="direction" style="margin: 0px; padding: 0px; border: 0px; font-style: inherit; font-variant: inherit; font-weight: inherit; font-stretch: inherit; line-height: inherit; font-family: inherit; vertical-align: baseline; background: none; color: rgb(107, 75, 161); hyphens: none;">direction</a> of an <a href="https://en.m.wiktionary.org/wiki/electric_current" title="electric current" style="margin: 0px; padding: 0px; border: 0px; font-style: inherit; font-variant: inherit; font-weight: inherit; font-stretch: inherit; line-height: inherit; font-family: inherit; vertical-align: baseline; background: none; color: rgb(107, 75, 161); hyphens: none;">electric current</a>.</li><li style="margin: 0px 0px 10px; padding: 0px; border: 0px; font-style: inherit; font-variant: inherit; font-weight: inherit; font-stretch: inherit; line-height: inherit; font-family: inherit; vertical-align: baseline; background: none;"><span class="ib-brac" style="margin: 0px; padding: 0px; border: 0px; font-style: inherit; font-variant: inherit; font-weight: inherit; font-stretch: inherit; line-height: inherit; font-family: inherit; vertical-align: baseline; background: none;">(<span class="ib-content" style="margin: 0px; padding: 0px; border: 0px; font-style: italic; font-variant: inherit; font-weight: inherit; font-stretch: inherit; line-height: inherit; font-family: inherit; vertical-align: baseline; background: none;"><a href="https://en.m.wiktionary.org/wiki/group_theory" title="group theory" style="margin: 0px; padding: 0px; border: 0px; font-style: inherit; font-variant: inherit; font-weight: inherit; font-stretch: inherit; line-height: inherit; font-family: inherit; vertical-align: baseline; background: none; color: rgb(107, 75, 161); hyphens: none;">group theory</a><span class="ib-brac" style="margin: 0px; padding: 0px; border: 0px; font-style: inherit; font-variant: inherit; font-weight: inherit; font-stretch: inherit; line-height: inherit; font-family: inherit; vertical-align: baseline; background: none;">) A binary map in a given <a href="https://en.m.wiktionary.org/wiki/group" title="group" style="margin: 0px; padding: 0px; border: 0px; font-style: inherit; font-variant: inherit; font-weight: inherit; font-stretch: inherit; line-height: inherit; font-family: inherit; vertical-align: baseline; background: none; color: rgb(107, 75, 161); hyphens: none;">group</a> <i style="margin: 0px; padding: 0px; border: 0px; font-variant: inherit; font-weight: inherit; font-stretch: inherit; line-height: inherit; font-family: inherit; vertical-align: baseline; background: none;">G, given by [<i style="margin: 0px; padding: 0px; border: 0px; font-variant: inherit; font-weight: inherit; font-stretch: inherit; line-height: inherit; font-family: inherit; vertical-align: baseline; background: none;">g, h] = <i style="margin: 0px; padding: 0px; border: 0px; font-variant: inherit; font-weight: inherit; font-stretch: inherit; line-height: inherit; font-family: inherit; vertical-align: baseline; background: none;">ghg<sup style="margin: 0px; padding: 0px; border: 0px; font-style: inherit; font-variant: inherit; font-weight: inherit; font-stretch: inherit; line-height: 1; font-family: inherit; font-size: 0.75em; background: none;">−1h<sup style="margin: 0px; padding: 0px; border: 0px; font-style: inherit; font-variant: inherit; font-weight: inherit; font-stretch: inherit; line-height: 1; font-family: inherit; font-size: 0.75em; background: none;">−1, where <i style="margin: 0px; padding: 0px; border: 0px; font-variant: inherit; font-weight: inherit; font-stretch: inherit; line-height: inherit; font-family: inherit; vertical-align: baseline; background: none;">g and <i style="margin: 0px; padding: 0px; border: 0px; font-variant: inherit; font-weight: inherit; font-stretch: inherit; line-height: inherit; font-family: inherit; vertical-align: baseline; background: none;">h are <a href="https://en.m.wiktionary.org/wiki/element" title="element" style="margin: 0px; padding: 0px; border: 0px; font-style: inherit; font-variant: inherit; font-weight: inherit; font-stretch: inherit; line-height: inherit; font-family: inherit; vertical-align: baseline; background: none; color: rgb(107, 75, 161); hyphens: none;">elements</a> of <i style="margin: 0px; padding: 0px; border: 0px; font-variant: inherit; font-weight: inherit; font-stretch: inherit; line-height: inherit; font-family: inherit; vertical-align: baseline; background: none;">G, which yields the group's <a href="https://en.m.wiktionary.org/wiki/identity" title="identity" style="margin: 0px; padding: 0px; border: 0px; font-style: inherit; font-variant: inherit; font-weight: inherit; font-stretch: inherit; line-height: inherit; font-family: inherit; vertical-align: baseline; background: none; color: rgb(107, 75, 161); hyphens: none;">identity</a> if and only if the group operation <a href="https://en.m.wiktionary.org/wiki/commute" title="commute" style="margin: 0px; padding: 0px; border: 0px; font-style: inherit; font-variant: inherit; font-weight: inherit; font-stretch: inherit; line-height: inherit; font-family: inherit; vertical-align: baseline; background: none; color: rgb(107, 75, 161); hyphens: none;">commutes</a> for <i style="margin: 0px; padding: 0px; border: 0px; font-variant: inherit; font-weight: inherit; font-stretch: inherit; line-height: inherit; font-family: inherit; vertical-align: baseline; background: none;">g and <i style="margin: 0px; padding: 0px; border: 0px; font-variant: inherit; font-weight: inherit; font-stretch: inherit; line-height: inherit; font-family: inherit; vertical-align: baseline; background: none;">h.</li><li style="margin-top: 0px; margin-right: 0px; margin-bottom: inherit; margin-left: 0px; padding: 0px; border: 0px; font-style: inherit; font-variant: inherit; font-weight: inherit; font-stretch: inherit; line-height: inherit; font-family: inherit; vertical-align: baseline; background: none;"><span class="ib-brac" style="margin: 0px; padding: 0px; border: 0px; font-style: inherit; font-variant: inherit; font-weight: inherit; font-stretch: inherit; line-height: inherit; font-family: inherit; vertical-align: baseline; background: none;">(<span class="ib-content" style="margin: 0px; padding: 0px; border: 0px; font-style: italic; font-variant: inherit; font-weight: inherit; font-stretch: inherit; line-height: inherit; font-family: inherit; vertical-align: baseline; background: none;">ring theory<span class="ib-brac" style="margin: 0px; padding: 0px; border: 0px; font-style: inherit; font-variant: inherit; font-weight: inherit; font-stretch: inherit; line-height: inherit; font-family: inherit; vertical-align: baseline; background: none;">) A binary map in a given <a href="https://en.m.wiktionary.org/wiki/ring" title="ring" style="margin: 0px; padding: 0px; border: 0px; font-style: inherit; font-variant: inherit; font-weight: inherit; font-stretch: inherit; line-height: inherit; font-family: inherit; vertical-align: baseline; background: none; color: rgb(107, 75, 161); hyphens: none;">ring</a> <i style="margin: 0px; padding: 0px; border: 0px; font-variant: inherit; font-weight: inherit; font-stretch: inherit; line-height: inherit; font-family: inherit; vertical-align: baseline; background: none;">R, given by [<i style="margin: 0px; padding: 0px; border: 0px; font-variant: inherit; font-weight: inherit; font-stretch: inherit; line-height: inherit; font-family: inherit; vertical-align: baseline; background: none;">a, b] = <i style="margin: 0px; padding: 0px; border: 0px; font-variant: inherit; font-weight: inherit; font-stretch: inherit; line-height: inherit; font-family: inherit; vertical-align: baseline; background: none;">ab − ba, where <i style="margin: 0px; padding: 0px; border: 0px; font-variant: inherit; font-weight: inherit; font-stretch: inherit; line-height: inherit; font-family: inherit; vertical-align: baseline; background: none;">a and <i style="margin: 0px; padding: 0px; border: 0px; font-variant: inherit; font-weight: inherit; font-stretch: inherit; line-height: inherit; font-family: inherit; vertical-align: baseline; background: none;">b are elements of <i style="margin: 0px; padding: 0px; border: 0px; font-variant: inherit; font-weight: inherit; font-stretch: inherit; line-height: inherit; font-family: inherit; vertical-align: baseline; background: none;">R, which yields the ring's <a href="https://en.m.wiktionary.org/wiki/zero" title="zero" style="margin: 0px; padding: 0px; border: 0px; font-style: inherit; font-variant: inherit; font-weight: inherit; font-stretch: inherit; line-height: inherit; font-family: inherit; vertical-align: baseline; background: none; color: rgb(107, 75, 161); hyphens: none;">zero</a> <a href="https://en.m.wiktionary.org/wiki/element" title="element" style="margin: 0px; padding: 0px; border: 0px; font-style: inherit; font-variant: inherit; font-weight: inherit; font-stretch: inherit; line-height: inherit; font-family: inherit; vertical-align: baseline; background: none; color: rgb(107, 75, 161); hyphens: none;">element</a> if and only if the multiplication operation commutes for <i style="margin: 0px; padding: 0px; border: 0px; font-variant: inherit; font-weight: inherit; font-stretch: inherit; line-height: inherit; font-family: inherit; vertical-align: baseline; background: none;">a and <i style="margin: 0px; padding: 0px; border: 0px; font-variant: inherit; font-weight: inherit; font-stretch: inherit; line-height: inherit; font-family: inherit; vertical-align: baseline; background: none;">b.</li></ol>