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Isolated subgroup

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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).


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References

[a1] A.G. Kurosh, "Theory of groups" , 2 , Chelsea (1960) pp. §66 (Translated from Russian)
How to Cite This Entry:
Isolated subgroup. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Isolated_subgroup&oldid=32631
This article was adapted from an original article by O.A. Ivanova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article