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