Isolated subgroup

A subgroup $A$ of a group $G$ such that $g\in A$ whenever $g^n\in A$, $g^n\ne 1$; in other words, if an equation $x^n=a$ (where $1\ne a\in A$) is solvable in $G$, then the solution lies in $A$. A subgroup $A$ is said to be strongly isolated if for every $a\in A$ the centralizer of $a$ in the whole group lies in $A$. The isolator of a set $M$ of elements of a group is the smallest isolated subgroup containing $M$.

In an $R$-group (that is, in a group with unique division), the concept of an isolated subgroup corresponds to that of a pure subgroup of an Abelian group. The intersection of isolated subgroups in an $R$-group is an isolated subgroup. A normal subgroup $H$ of an $R$-group $G$ is isolated if and only if the quotient group $G/H$ is torsion-free. The centre of an $R$-group is isolated.

In the theory of ordered groups, isolated subgroups are sometimes referred to as convex subgroups (cf. Convex subgroup).

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