Torsion subgroup of an abelian group

Given an Abelian group $𝐺$ we may form the torsion subgroup $𝐺 𝑇$ containing all elements of finite order, i.e. $𝐺 𝑇 = {𝑥 \in 𝐺 ∣ \exists 𝑛 \in ℕ 𝑥 𝑛 = 𝑒}$ . #m/def/group #m/thm/group

Proof of subgroup

Clearly $𝑒 \in 𝐺 𝑇$ , so the set is inhabited. Let $𝑎, 𝑏 \in 𝐺 𝑇$ , so that there exist $𝑚, 𝑛 \in ℕ$ such that $𝑎 𝑚 = 𝑏 𝑛 = 𝑒$ . Then $(𝑎 𝑏 - 1) 𝑚 𝑛 = 𝑎 𝑚 𝑛 𝑏 - 𝑚 𝑛 = (𝑎 𝑚) 𝑛 (𝑏 𝑛) - 𝑚 = 𝑒$ , hence $𝑎 𝑏 - 1 \in 𝐺 𝑇$ . Therefore $𝐺 𝑇$ is a subgroup by One step subgroup test.

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