Conjugacy Action is Group Action

From ProofWiki
Jump to navigation Jump to search

Theorem

Let $\left({G, \circ}\right)$ be a group whose identity is $e$.


Conjugacy Action on Group Elements is Group Action

Let $\struct {G, \circ}$ be a group whose identity is $e$.


The conjugacy action on $G$:

$\forall g, h \in G: g * h = g \circ h \circ g^{-1}$

is a group action on itself.


Conjugacy Action on Subgroups is Group Action

Let $X$ be the set of all subgroups of $G$.

For any $H \le G$ and for any $g \in G$, the conjugacy action:

$g * H := g \circ H \circ g^{-1}$

is a group action.


Conjugacy Action on Subsets is Group Action

Let $\powerset G$ be the set of all subgroups of $G$.

For any $S \in \powerset G$ and for any $g \in G$, the conjugacy action:

$g * S := g \circ S \circ g^{-1}$

is a group action.