Conjugate of Subgroup is Subgroup/Proof 2
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Theorem
Let $G$ be a group.
Let $H \le G$ be a subgroup of $G$.
Then the conjugate of $H$ by $a$ is a subgroup of $G$:
- $\forall H \le G, a \in G: H^a \le G$
Proof
Let $*: G \times G / H \to G / H$ be the group action on the (left) coset space:
- $\forall g \in G, \forall g' H \in G / H: g * \paren {g' H} := \paren {g g'} H$
It is established in Action of Group on Coset Space is Group Action that $*$ is a group action.
Then from Stabilizer of Coset under Group Action on Coset Space:
- $\Stab {a H} = a H a^{-1}$
where $\Stab {a H}$ the stabilizer of $a H$ under $*$.
The result follows from Stabilizer is Subgroup.
$\blacksquare$
Sources
- 1965: J.A. Green: Sets and Groups ... (previous) ... (next): Chapter $6$: Cosets: Exercise $9$