Definition:Subgroup Action

Definition

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

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

Let $*: H \times G \to G$ be the operation defined as:

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

This is the subgroup action of $H$ on $G$.

Also see

• Subgroup Action is Group Action

• Results about the subgroup action can be found here.

Sources

• 1965: J.A. Green: Sets and Groups ... (previous) ... (next): $\S 5.5$. Groups acting on sets: Example $106$
• 1971: Allan Clark: Elements of Abstract Algebra ... (previous) ... (next): Chapter $2$: The Sylow Theorems: $\S 54 \alpha$