Pure subgroup

In mathematics, especially in the area of algebra studying the theory of abelian groups, a pure subgroup is a generalization of direct summand. It has found many uses in abelian group theory and related areas.

A subgroup ${\displaystyle S}$ of a (typically abelian) group ${\displaystyle G}$ is said to be pure if whenever an element of ${\displaystyle S}$ has an ${\displaystyle n^{th}}$ root in ${\displaystyle G}$, it necessarily has an ${\displaystyle n^{th}}$ root in ${\displaystyle S}$. Formally, ${\displaystyle \forall n\in \mathbb {Z} ,a\in S,x^{n}=a}$ is solvable in ${\displaystyle G}$ ${\displaystyle \Rightarrow x^{n}=a}$ is solvable in ${\displaystyle S}$.

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