Answer to Question #242447 in Abstract Algebra for Eshetie

Question

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�

Answer 1

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

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

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

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

Also,

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

This shows that A is a subgroup of G.