## Groups whose automorphisms are almost determined by their restriction to a subgroup.(English)Zbl 0581.20042

A group is said to be torsion separable if it has a series of finite length whose factors either are periodic or have no non-trivial periodic quotients. The main result is: Theorem. Let G be a torsion separable group and let H be a Chernikov subgroup with only finitely many conjugates. Assume that $$C_{Aut G}(H)$$ is countable and $$C_{Inn G}(H)$$ is Chernikov. Then G is a Chernikov group and $$| G:H^ G|$$ is finite.
Reviewer: D.J.S.Robinson

### MSC:

 20F28 Automorphism groups of groups 20E07 Subgroup theorems; subgroup growth 20E15 Chains and lattices of subgroups, subnormal subgroups 20E36 Automorphisms of infinite groups
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### References:

 [1] Robinson, A Course in the Theory of Groups (1982) · Zbl 0483.20001 [2] DOI: 10.1112/plms/s3-35.1.34 · Zbl 0358.20052 [3] Robinson, Symposia Mathematica, Vol. XVII pp 441– (1976) [4] DOI: 10.2307/1993045 · Zbl 0065.01001 [5] Groot, Nederl. Akad. Wetensch. Proc. Ser. A 19 pp 137– (1957) [6] DOI: 10.1007/BF02024508 · Zbl 0093.02901 [7] DOI: 10.1016/0021-8693(69)90052-0 · Zbl 0214.05606 [8] Robinson, Finiteness Conditions and Generalized Soluble Groups I (1972)
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