# zbMATH — the first resource for mathematics

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
Full Text:
##### References:
 [1] V. P. Shunkov, ?On periodic groups with an almost regular involution,? Algebra i Logika,11, No. 4, 470-493 (1972). [2] B. Hartley and Th. Meixner, ?Periodic groups in which the centralizer of an involution has bounded order,? J. Algebra,64, No. 1, 285-291 (1980). · Zbl 0429.20039 [3] V. V. Belyaev and N. F. Sesekin, ?Periodic groups with an almost regular involutive automorphism,? Stud. Sci. Math. Hung.,17, Nos. 1-4, 137-141 (1982). · Zbl 0555.20027 [4] A. O. Asar, ?Locally finite groups with Cernikov centralizers,? J. Algebra,68, No. 1, 170-176 (1981). · Zbl 0451.20036 [5] A. O. Asar, ?The solution of a problem of Kegel and Wehrfritz,? Proc. London Math. Soc.,45, No. 2, 337-364 (1982). · Zbl 0498.20027 [6] I. I. Pavlyuk, ?On a problem of Kegel and Wehrfritz,? in: The 7th All-Union Symp. on Group Theory, Proceedings [in Russian], Krasnoyarsk (1980), p. 83. [7] B. Hartley, ?Periodic locally soluble groups containing an element of prime order with Cernikov centralizer,? Quart. J. Math.,33, No. 131, 309-323 (1982). · Zbl 0495.20011 [8] D. Gorenstein, Finite Groups, Harper & Row, New York (1968). [9] O. Kegel and B. A. F. Wehrfritz, Locally Finite Groups, Amsterdam-London (1973). · Zbl 0259.20001 [10] M. I. Kargapolov and Yu. I. Merzlyakov, Fundamentals of Group Theory [in Russian], 3rd ed., Nauka, Moscow (1982). · Zbl 0508.20001 [11] D. J. S. Robinson, Finiteness Conditions and Generalized Soluble Groups. Part I, Springer-Verlag, New York (1972). · Zbl 0243.20032
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.