Torsion subgroup

Definition

Let $A$ be an abelian group. The torsion subgroup of $A$ is $A_T = \{a \in A : a \text{ has finite order}\}$. The group $A$ is a torsion group when $A_T = A$, and it is torsion-free when $A_T = \{e\}$.

Remarks

The set $A_T$ is a subgroup because $A$ is commutative: if $a^m = e$ and $b^n = e$, then $(a b)^{m n} = e$. In a nonabelian group, the elements of finite order need not form a subgroup. In $\langle x, y \mid x^2, y^2 \rangle$, the product $x y$ has infinite order. The quotient group $A / A_T$ is torsion-free, and $A_T$ is a characteristic subgroup of $A$. For $A = \mathbb{Q}/\mathbb{Z}$, each element has finite order, so $A_T = A$, but the orders have no upper bound.

Explanations

L1 human

The torsion subgroup collects the elements that return to the identity after finitely many steps. In an abelian group, these elements form a subgroup. Removing them leaves a group in which only the identity has finite order.