Let $G$ be a group, let $V$ be a finite-dimensional [vector space](/page/Vector%20Space) over $\mathbb{C}$, and let \begin{align*} \rho: G \to GL(V) \end{align*} be a group representation. Define the character of $\rho$ to be the map \begin{align*} \chi_\rho: G &\to \mathbb{C} \\ g &\mapsto \operatorname{tr}(\rho(g)). \end{align*} Then $\chi_\rho$ is a class function; equivalently, for every $g,h \in G$, \begin{align*} \chi_\rho(hgh^{-1}) = \chi_\rho(g). \end{align*}

Let $G$ be a finite group, and let $\operatorname{Cl}(G)$ denote the set of conjugacy classes of $G$. Define the complex [vector space](/page/Vector%20Space) of class functions on $G$ by \begin{align*} \operatorname{CF}(G) := \{f: G \to \mathbb{C} \mid f(hgh^{-1}) = f(g) \text{ for all } g,h \in G\}. \end{align*} Then \begin{align*} \dim_{\mathbb{C}} \operatorname{CF}(G) = |\operatorname{Cl}(G)|. \end{align*}

Let $G$ be a group with operation written multiplicatively, and let $A \subset G$. For each $g \in G$, define the [conjugacy class](/page/Conjugacy%20Class) of $g$ in $G$ by \begin{align*} \operatorname{Cl}_G(g) := \{hgh^{-1} : h \in G\}. \end{align*} Then $A$ is conjugation-invariant, meaning that \begin{align*} \forall a \in A,\ \forall h \in G,\quad hah^{-1} \in A, \end{align*} if and only if $A$ is a union of conjugacy classes of $G$, meaning that there exists a subset $S \subset G$ such that \begin{align*} A = \bigcup_{s \in S} \operatorname{Cl}_G(s). \end{align*}

Let $G$ be a group with identity element $e$, let $X$ be a set, and let \begin{align*} \alpha: G \times X &\to X \\ (g, x) &\mapsto g \cdot x \end{align*} be a left action of $G$ on $X$. Let $\operatorname{Sym}(X)$ denote the group of bijections $X \to X$ under composition. Then the map \begin{align*} \rho: G &\to \operatorname{Sym}(X) \\ g &\mapsto \rho(g) \end{align*} defined by \begin{align*} \rho(g): X &\to X \\ x &\mapsto g \cdot x \end{align*} is a [group homomorphism](/page/Group%20Homomorphism). Moreover, \begin{align*} \ker \rho = \{g \in G : g \cdot x = x \text{ for every } x \in X\}, \end{align*} so the action is faithful if and only if $\rho$ is injective.

Let $G$ be a Lie group acting smoothly and transitively on a smooth manifold $M$, and let $p \in M$. Suppose $G_p$ is a closed Lie subgroup and $G/G_p$ is equipped with its quotient smooth manifold structure. Then the map \begin{align*} G/G_p &\to M \\ gG_p &\mapsto g \cdot p \end{align*} is a $G$-equivariant diffeomorphism.

Let $G$ be a group with identity element $e$, let $X$ be a set, and let \begin{align*} \alpha: G \times X &\to X \\ (g, y) &\mapsto g \cdot y \end{align*} be a left [group action](/page/Group%20Action) of $G$ on $X$. For each $x \in X$, the stabiliser \begin{align*} G_x := \{g \in G : g \cdot x = x\} \end{align*} is a subgroup of $G$.

Let $G$ be a group acting on a set $X$ by the action map \begin{align*} G \times X &\to X \\ (g, x) &\mapsto g \cdot x. \end{align*} For each $x \in X$, let \begin{align*} G_x := \{h \in G : h \cdot x = x\} \end{align*} denote the stabilizer of $x$. Then, for every $g \in G$ and every $x \in X$, \begin{align*} G_{g \cdot x} = gG_xg^{-1}. \end{align*}

Let $G$ be a group acting on a set $X$ by a map \begin{align*} G \times X &\to X \\ (g, x) &\mapsto g \cdot x. \end{align*} For each $x \in X$, define the orbit of $x$ by \begin{align*} G \cdot x := \{g \cdot x : g \in G\}. \end{align*} Then the collection of orbits \begin{align*} X/G := \{G \cdot x : x \in X\} \end{align*} is a partition of $X$: every element of $X$ lies in at least one orbit, and any two orbits are either equal or disjoint.

Let $(X,\tau)$ be a [topological space](/page/Topological%20Space). Then $(X,\tau)$ is connected if and only if every subset $A \subset X$ that is both open and closed in $(X,\tau)$ satisfies $A = \varnothing$ or $A = X$.

Let $(X,\tau_X)$ and $(Y,\tau_Y)$ be topological spaces, let $f: X \to Y$ be continuous, and let $C \subset X$ be a connected component of $X$. Then there exists a connected component $D \subset Y$ such that $f(C) \subset D$.