Let A={a∈G:O(a)∣n}
We shall show that A is a subgroup of G using two steps subgroup test.\\
Let a,b∈A⟹a,b∈G:O(a)∣n and O(b)∣n \\
⟹ab∈GO(ab)=lcm(O(a),O(b)){Since G is an Abelian group}⟹O(ab)∣n⟹ab∈A
Also,
a−1∈G since a∈GO(a−1)=O(a){Element of a group and its inverse have same order}⟹O(a−1)∣n
This shows that A is a subgroup of G.
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