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On groups in which the centralizer of an involution satisfies min-\(\pi\). (English. Russian original) Zbl 0795.20020
Sib. Math. J. 31, No. 5, 864-867 (1990); translation from Sib. Mat. Zh. 31, No. 5(183), 197-200 (1990).
The following theorem was proved by the reviewer [Q. J. Math., Oxf. II. Ser. 33, 309-323 (1982; Zbl 0495.20011)]. Let \(G\) be a periodic almost locally soluble group containing an involution \(\varphi\) whose centralizer is a Chernikov group. Then \(G/[G,\varphi]\) and \([G,\varphi]'\) are Chernikov groups. The main theorem here has a similar flavour. Theorem: Let \(\pi\) be a finite set of primes containing 2, and let \(G\) be a periodic almost locally soluble group containing an involution whose centralizer satisfies \(\text{Min-}\pi\). Suppose that \([G,\varphi]\) is almost soluble. Then the commutator subgroup \([G,\varphi]'\) satisfies \(\text{Min-}\pi\). The proof is quite intricate. The assumption that \([G,\varphi]\) is almost soluble enables induction to be used on the derived length of a subgroup of finite index in this group.
The author has since considerably improved this result, by removing the hypothesis that \([G,\varphi]\) is almost soluble and obtaining the further conclusion that \(G/[G,\varphi]\) satisfies \(\text{Min-}\pi\) [J. Algebra 155, 36-43 (1993; Zbl 0780.20024)]. The methods are quite different.
MSC:
20F50 Periodic groups; locally finite groups
20F28 Automorphism groups of groups
20E36 Automorphisms of infinite groups
20E25 Local properties of groups
20F19 Generalizations of solvable and nilpotent groups
20E15 Chains and lattices of subgroups, subnormal subgroups
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References:
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