Subgroup Action is Group Action

From ProofWiki
Jump to navigation Jump to search


It has been suggested that this page or section be merged into Group Action on Subgroup by Left Regular Representation.
To discuss this page in more detail, feel free to use the talk page.
When this work has been completed, you may remove this instance of {{Mergeto}} from the code.


Theorem

Let $\struct {G, \circ}$ be a group.

Let $\struct {H, \circ}$ be a subgroup of $G$.

Let $*: H \times G \to G$ be the subgroup action defined for all $h \in H, g \in G$ as:

$\forall h \in H, g \in G: h * g := h \circ g$


Then $*$ is a group action.


Proof

Let $g \in G$.

First we note that since $G$ is closed, and $h \circ g \in G$, it follows that $h * g \in G$.


Next we note:

$e * g = e \circ g = g$

and so Group Action Axiom $\text {GA} 2$ is satisfied.


Now let $h_1, h_2 \in G$.

We have:

\(\ds \paren {h_1 \circ h_2} * g\) \(=\) \(\ds \paren {h_1 \circ h_2} \circ g\) Definition of $*$
\(\ds \) \(=\) \(\ds h_1 \circ \paren {h_2 \circ g}\) Group Axiom $\text G 1$: Associativity
\(\ds \) \(=\) \(\ds h_1 * \paren {h_2 * g}\) Definition of $*$

and so Group Action Axiom $\text {GA} 1$ is satisfied.

Hence the result.

$\blacksquare$


Sources

Retrieved from "https://proofwiki.org/w/index.php?title=Subgroup_Action_is_Group_Action&oldid=377953"