Subset Product Action is Group Action

Theorem

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

Let $\powerset G$ be the power set of $\struct {G, \circ}$.

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

$\forall g \in G: \forall S \in \powerset G: g * S = g \circ S$

is a group action.

Let $g \in G$.

First we note that since $G$ is closed, and $g \circ S$ consists of products of elements of $G$, it follows that:

$g * S \subseteq G$

Next we note:

$e * S = e \circ S = \set {e \circ s: s \in S} = \set {s: s \in S} = S$

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

Now let $g, h \in G$.

We have:

$\ds \paren {g \circ h} * S$

$=$

$\ds \paren {g \circ h} \circ S$

Definition of Subset Product Action

$\ds $

$=$

$\ds \set {\paren {g \circ h} \circ s: s \in S}$

$\ds $

$=$

$\ds \set {g \circ \paren {h \circ s}: s \in S}$

Group Axiom $\text G 1$: Associativity

$\ds $

$=$

$\ds g * \set {h \circ s: s \in S}$

$\ds $

$=$

$\ds g * \paren {h \circ S}$

$\ds $

$=$

$\ds g * \paren {h * S}$

Definition of Subset Product Action

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

Hence the result.

$\blacksquare$

1965: J.A. Green: Sets and Groups ... (previous) ... (next): $\S 5.5$. Groups acting on sets: Example $104$
1982: P.M. Cohn: Algebra Volume 1 (2nd ed.) ... (previous) ... (next): $\S 3.3$: Group actions and coset decompositions
1996: John F. Humphreys: A Course in Group Theory ... (previous) ... (next): Chapter $10$: The Orbit-Stabiliser Theorem: Example $10.6$