Conjugate Elements

2020 Mathematics Subject Classification: Primary: 20-XX [MSN][ZBL] in a group $G$

Elements $x$ and $x'$ of $G$ for which $$ x' = g^{-1} x g $$ for some $g$ in $G$. One also says that $x'$ is the result of conjugating $x$ by $g$. The power notation $x^g$ is frequently used for the conjugate of $x$ under $g$.

Let $A,B$ be two subsets of a group $G$, then $A^B$ denotes the set $$ \{ a^b : a \in A\,,\, b \in B \} $$ For some fixed $g$ in $G$ and some subset $M$ of $G$ the set $M^g = \{ m^g : m \in M\}$ is said to be conjugate to the set $M$ in $G$. In particular, two subgroups $U$ and $V$ are called conjugate subgroups if $V = U^g$ for some $g$ in $G$. If a subgroup $H$ coincides with $H^g$ for every $g \in G$ (that is, $H$ consists of all conjugates of all its elements), then $H$ is called a normal subgroup of $G$ (or an invariant subgroup, or, rarely, a self-conjugate subgroup).

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Conjugacy of elements is an equivalence relation on $G$, and the equivalence classes are the conjugacy classes of $G$.

The map $x \mapsto g^{-1} x g$ for given $g$ is conjugation by $g$: it is an inner automorphism of $G$.

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