A subgroup of a group is said to be a normal subgroup of if for all
and .
We use the notation to signify that is a normal subgroup of .
Claim (Lemma)
Every subgroup of an abelian group is a normal subgroup
Proof
Let be a subgroup of an abelian group .
Then,
.
So, for all and , where is the identiy element, the result holds.
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