Subset Product Action is Group Action
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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.
Proof
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:
\(\ds \paren {g \circ h} * S\) | \(=\) | \(\ds \paren {g \circ h} \circ S\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \set {\paren {g \circ h} \circ s: s \in S}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \set {g \circ \paren {h \circ s}: s \in S}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds g * \set {h \circ s: s \in S}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds g * \paren {h \circ S}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds g * \paren {h * S}\) |
and so Group Action Axiom $\text {GA} 1$ is satisfied.
Hence the result.
$\blacksquare$
Also see
- Stabilizer of Subset Product Action on Power Set
- Stabilizer of Coset Action on Set of Subgroups
- Orbit of Subgroup under Coset Action is Coset Space
Sources
- 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 $10.6$