Standard Subsets
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\({\mathcal{P}}(X), 2^X \coloneqq\mathop{\mathrm{Hom}}_{\mathsf{Set}}(\left\{{0, 1}\right\}, X)\) all define the powerset of \(X\).
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Commuting: \begin{align*} [g, h], [gh] && \text{Commutator of elements} \\ && \coloneqq ghg^{-1}\in G \\ \\ [G,H], [GH] && \text{Commutator of subgroups} \\ && \coloneqq\left\langle{ \left\{{ [gh] {~\mathrel{\Big\vert}~}g \in G,\, h \in H }\right\} }\right\rangle \leq G .\end{align*}
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Central: \begin{align*} Z(G) && \text{Center of a group} \\ &&\coloneqq\left\{{ x\in G {~\mathrel{\Big\vert}~}\forall g\in G,\, gxg ^{-1} = x }\right\} \subseteq G \\ \\ C_G(x), Z(x) && \text{Centralizer of an element} \\ && \coloneqq\left\{{g\in G {~\mathrel{\Big\vert}~}[g, x] = 1}\right\} \subseteq G \\ \\ C_G(H), Z_G(H) && \text{Centralizer of a subgroup} \\ && \coloneqq\left\{{g\in G {~\mathrel{\Big\vert}~}[g, x] = 1\,\, \forall h\in H}\right\} = \bigcap_{h\in H} C_H(h) \subseteq G \\ \\ [h], {\mathrm{Conj}}(h) && \text{Conjuacy class of an element} \\ && \coloneqq\left\{{ ghg ^{-1} {~\mathrel{\Big\vert}~}g\in G}\right\} \leq G \subseteq G \\ \\ .\end{align*}
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Normal: \begin{align*} N_G(H) && \text{Normalizer of a subgroup} \\ && \coloneqq\left\{{ g\in G {~\mathrel{\Big\vert}~}gHg ^{-1} = H }\right\} \subseteq G .\end{align*}
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Automorphisms: \begin{align*} \mathop{\mathrm{Inn}}(G) && \text{Inner automorphisms} \\ && \coloneqq\left\{{ \varphi _g(x) \coloneqq gxg ^{-1} }\right\} \subseteq \mathop{\mathrm{Aut}}(G) \\ \\ \mathop{\mathrm{Out}}(G) && \text{Outer automorphisms}\\ && \coloneqq\mathop{\mathrm{Aut}}(G) / \mathop{\mathrm{Inn}}(G) \leftarrow\mathop{\mathrm{Aut}}(G) .\end{align*}
Group Actions
\begin{align*} g\cdot x, g\curvearrowright x && \text{The action of a group on a set} \\ && \coloneqq\phi(g)(x) \text{ where } \phi: G\to \mathop{\mathrm{Aut}}_{\mathsf{Set}}(X) \in {\mathsf{Grp}}\\ \\ {\mathrm{Orb}}(x), Gx && \text{Orbit} \\ && \coloneqq\left\{{g\cdot x {~\mathrel{\Big\vert}~}g\in G}\right\} \subseteq X \\ \\ {\operatorname{Stab}}(x), {\operatorname{Stab}}_G(x), G_x && \text{Stabilizer} \\ && \coloneqq\left\{{ g \in G {~\mathrel{\Big\vert}~}g\cdot x=x }\right\} \leq G \\ \\ X/G && \text{Set of orbits} \\ && \coloneqq\left\{{{\mathrm{Orb}}(x) {~\mathrel{\Big\vert}~}x\in X}\right\} \subseteq 2^X \\ \\ \mathrm{Fix}(X), \mathrm{Fix}_G(X), X^G && \text{Set of fixed points}\\ && \coloneqq\left\{{x\in X {~\mathrel{\Big\vert}~}g\cdot x = x\, \forall g\in G}\right\} \subseteq X .\end{align*}