Conjugate of Subgroup is Subgroup
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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 1
Let $H \le G$.
First, we show that $x, y \in H^a \implies x \circ y \in H^a$:
\(\ds x, y\) | \(\in\) | \(\ds H^a\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds a x a^{-1}, a y a^{-1}\) | \(\in\) | \(\ds H\) | Definition of Conjugate of Group Subset | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \paren {a x a^{-1} } \paren {a y a^{-1} }\) | \(\in\) | \(\ds H\) | Group Axiom $\text G 0$: Closure | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds a \paren {x y} a^{-1}\) | \(\in\) | \(\ds H\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds x y\) | \(\in\) | \(\ds H^a\) | Definition of Conjugate of Group Subset |
Next, we show that $x \in H^a \implies x^{-1} \in H^a$:
\(\ds x\) | \(\in\) | \(\ds H^a\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds a x a^{-1}\) | \(\in\) | \(\ds H\) | Definition of Conjugate of Group Subset | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \paren {a x a^{-1} }^{-1} = a x^{-1} a^{-1}\) | \(\in\) | \(\ds H\) | Power of Conjugate equals Conjugate of Power | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds x^{-1}\) | \(\in\) | \(\ds H^a\) | Definition of Conjugate of Group Subset |
Thus by the Two-Step Subgroup Test, $H^a \le G$.
$\blacksquare$
Proof 2
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
- 1967: George McCarty: Topology: An Introduction with Application to Topological Groups ... (previous) ... (next): Chapter $\text{II}$: Groups: Exercise $\text{R}$
- 1971: Allan Clark: Elements of Abstract Algebra ... (previous) ... (next): Chapter $2$: Subgroups and Cosets: $\S 35 \gamma$
- 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): Chapter $7$: Cosets and Lagrange's Theorem: Exercise $5$
- 1996: John F. Humphreys: A Course in Group Theory ... (previous) ... (next): Chapter $7$: Normal subgroups and quotient groups: Exercise $1$