Question #121389
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
1
Expert's answer
2020-06-10T19:12:27-0400

Given that,GG is finite group and HH is subgroup of GG

Claim: If HGH\triangleleft G then, for any aGa\in G ,aH=HaaH=Ha

Proof:

Suppose, HG    aHa1HH\triangleleft G\implies aHa^{-1}\subset H

Note that H=a(a1H(a1)1)a1=a(a1Ha)a1aHa1H=a(a^{-1}H(a^{-1})^{-1})a^{-1}=a(a^{-1}Ha)a^{-1}\subset aHa^{-1}

Hence, from above two equation aHa1=H    aH=HaaHa^{-1}=H\implies aH=Ha


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