Question #242447
Let G be an abelian group and n be a positive integer. Then prove that the set
{a ∈ G : θ(a) divides n} is a subgroup of G�
1
Expert's answer
2021-09-27T16:34:38-0400

Let A={aG:O(a)n}A=\{a\in G : O(a)|n\}\\

We shall show that A is a subgroup of G using two steps subgroup test.\\

Let a,bA    a,bG:O(a)n and O(b)na,b \in A \implies a,b\in G: O(a)|n \text{ and } O(b)|n \\

    abGO(ab)=lcm(O(a),O(b)){Since G is an Abelian group}    O(ab)n    abA\implies ab \in G\\ O(ab)=lcm(O(a),O(b)) \{\text{Since G is an Abelian group}\}\\ \implies O(ab)|n\\ \implies ab\in A

Also,

a1G since aGO(a1)=O(a){Element of a group and its inverse have same order}    O(a1)na^{-1}\in G \text{ since } a \in G\\ O(a^{-1})=O(a) \{\text{Element of a group and its inverse have same order}\}\\ \implies O(a^{-1})|n

This shows that A is a subgroup of G.


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