Given that,GGG is finite group and HHH is subgroup of GGG
Claim: If H◃GH\triangleleft GH◃G then, for any a∈Ga\in Ga∈G ,aH=HaaH=HaaH=Ha
Proof:
Suppose, H◃G ⟹ aHa−1⊂HH\triangleleft G\implies aHa^{-1}\subset HH◃G⟹aHa−1⊂H
Note that H=a(a−1H(a−1)−1)a−1=a(a−1Ha)a−1⊂aHa−1H=a(a^{-1}H(a^{-1})^{-1})a^{-1}=a(a^{-1}Ha)a^{-1}\subset aHa^{-1}H=a(a−1H(a−1)−1)a−1=a(a−1Ha)a−1⊂aHa−1
Hence, from above two equation aHa−1=H ⟹ aH=HaaHa^{-1}=H\implies aH=HaaHa−1=H⟹aH=Ha
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