Isolated subgroup

A subgroup of a group such that whenever , ; in other words, if an equation (where ) is solvable in , then the solution lies in . A subgroup is said to be strongly isolated if for every the centralizer of in the whole group lies in . The isolator of a set of elements of a group is the smallest isolated subgroup containing .

In an -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 -group is an isolated subgroup. A normal subgroup of an -group is isolated if and only if the quotient group is torsion-free. The centre of an -group is isolated.

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

References

[a1]	A.G. Kurosh, "Theory of groups" , 2 , Chelsea (1960) pp. §66 (Translated from Russian)