Groups with given properties of finite subgroups. (English. Russian original) Zbl 1266.20047

Lytkina, D. V.; Mazurov, V. D.

Algebra Logic 51, No. 3, 213-219 (2012); translation from Algebra Logika 51, No. 3, 321-330 (2012).

Summary: Suppose that in every finite even order subgroup \(F\) of a periodic group \(G\), the equality \([u,x]^2=1\) holds for any involution \(u\) of \(F\) and for an arbitrary element \(x\) of \(F\). Then the subgroup \(I\) generated by all involutions in \(G\) is locally finite and is a 2-group. In addition, the normal closure of every subgroup of order 2 in \(G\) is commutative.

MSC:

20F50	Periodic groups; locally finite groups
20E07	Subgroup theorems; subgroup growth
20E25	Local properties of groups
20F05	Generators, relations, and presentations of groups

Keywords:

periodic groups; finite subgroups; subgroups of even order; locally finite groups; involutions; normal closures

Software:

GAP

References:

[1]	S. I. Adyan, ”On subgroups of free periodic groups of odd exponent,” Tr. Mat. Inst. Akad. Nauk SSSR, 112, 64-72 (1971).
[2]	S. I. Adyan, The Burnside Problem and Identities in Groups [in Russian], Nauka, Moscow (1975). · Zbl 0306.20045
[3]	D. V. Lytkina, ”2-Groups with given properties of finite subgroups,” Vlad. Mat. Zh., 13, No. 4, 35-39 (2011). · Zbl 1275.20040
[4]	I. N. Sanov, ”Solving the Burnside problem for period 4,” Uch. Zap. LGU, Ser. Mat., 10, 166-170 (1940). · Zbl 0061.02506
[5]	The GAP Group, GAP–Groups, Algorithms, and Programming, Vers. 4.4.12 (2008); http://www.gap-system.org .
[6]	G. Havas and C. Ramsay, ”On proofs in finitely presented groups,” London Math. Soc. Lect. Note Ser., 340, Cambridge Univ. Press, Cambridge (2007), pp. 457-474. · Zbl 1120.20032
[7]	I. D. Macdonald, ”On certain varieties of groups,” Math. Z., 76, 270-282 (1961). · Zbl 0104.02202