Square map is endomorphism iff abelian
This article describes an easy-to-prove fact about basic notions in group theory, that is not very well-known or important in itself
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Let be a group and be the map defined as . Then, is an endomorphism if and only if is Abelian.
From endomorphism to Abelian
Suppose is an endomorphism of the group . Then for any we want to show that and commute. This can be proved as follows:
becaus is an endomorphism
Cancelling the leftmost and the rightmost , we get:
and hence commute.