Subgroup Action is Group Action

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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.


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.