Fixed order subgroup of an abelian group

Given an Abelian group , we may construct a subgroup containing only those elements whose order divides a given integer , i.e. . #m/thm/group

Proof

Let Clearly so is inhabited. Let . Since is abelian , it follows . Therefore is a subgroup by One step subgroup test.

This construction can fail for non-abelian groups, for example in the set is not closed.

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