Stabilizer of Subgroup Action on Left Coset Space

Theorem

Let $G$ be a group.

Let $H$ and $K$ be subgroups of $G$.

Let $K$ act on the left coset space $G / H^l$ by:

$\forall \tuple {k, g H} \in K \times G / H^l: k * g H := \paren {k g} H$

The stabilizer of $g H$ is $K \cap H^g$, where $H^g$ denotes the $G$-conjugate of $H$ by $g$.

Proof

\(\ds \Stab {g H}\)

\(=\)

\(\ds \set {k \in K: \paren {k g} H = g H}\)

Definition of Stabilizer

\(\ds \)

\(=\)

\(\ds \set {k \in K: g^{-1} \paren {k g} \in H}\)

Left Cosets are Equal iff Product with Inverse in Subgroup

\(\ds \)

\(=\)

\(\ds \set {k \in K: k \in g H g^{-1} }\)

\(\ds \)

\(=\)

\(\ds K \cap H^g\)

Definition of Conjugate of Group Subset

$\blacksquare$

Sources

• 1971: Allan Clark: Elements of Abstract Algebra ... (previous) ... (next): Chapter $2$: The Sylow Theorems: $\S 54 \gamma$