Homework SourseWhat is the stabilizer of the coset aH for the operation of
What is the stabilizer of the coset [aH] for the operation of G on G / H?
Solution
The stabilizer of aH is the conjugate subgroup aHa-1.
Proof:
By definition, the stabilizer of aH is GaH={g in G : g(aH)=aH}.
But gaH=aH ó a-1gaH=H (via left multiplication of both sides by a-1 to get “=>”, or by a to get , “<=”),
and a-1gaH=H ó a-1ga in H ó g in aHa-1.
![What is the stabilizer of the coset [aH] for the operation of G on G / H?SolutionThe stabilizer of aH is the conjugate subgroup aHa-1. Proof: By definition, the](/WebImages/6/what-is-the-stabilizer-of-the-coset-ah-for-the-operation-of-989142-1761508436-0.webp "What is the stabilizer of the coset [aH] for the operation of G on G / H?SolutionThe stabilizer of aH is the conjugate subgroup aHa-1. Proof: By definition, the")
