Answer to Question #121389 in Abstract Algebra for Samson

Question

If H is a finite subgroup of a group G, for all
a
ϵ
G
aϵG the left coset aH and the right coset Ha coincide if

Answer 1

Given that,$G$ is finite group and $H$ is subgroup of $G$

Claim: If $H\triangleleft G$ then, for any $a\in G$ ,$aH=Ha$

Proof:

Suppose, $H\triangleleft G\implies aHa^{-1}\subset H$

Note that $H=a(a^{-1}H(a^{-1})^{-1})a^{-1}=a(a^{-1}Ha)a^{-1}\subset aHa^{-1}$

Hence, from above two equation $aHa^{-1}=H\implies aH=Ha$