On solvable groups of exponent \(4\). (Russian, English) Zbl 1034.20031

Deryabina, G. S.; Krasil’nikov, A. N.

Sib. Mat. Zh. 44, No. 1, 69-72 (2003); translation in Sib. Math. J. 44, No. 1, 58-60 (2003).

The authors give a negative answer to V. V. Bludov’s question 13.10 from the Kourovka notebook [2002; Zbl 0999.20001]: Does there exist a function of the natural numbers \(f\colon\mathbb{N}\to\mathbb{N}\) with the property that, for any group \(G\) of solvability length \(k\) and with generating set \(A\), if the identity \(x^4=1\) is satisfied in any subgroup generated by less than \(f(k)\) elements then this identity is satisfied in the whole group \(G\).
More precisely, Theorem 1 gives a positive answer to the problem for \(k=2\) while in Theorem 2 the authors give a negative answer to the problem for \(k=3\).

Reviewer: K. N. Ponomarev (Novosibirsk)

MSC:

20F16	Solvable groups, supersolvable groups
20F50	Periodic groups; locally finite groups
20E25	Local properties of groups
20F05	Generators, relations, and presentations of groups

Keywords:

identities; metabelian groups; solvable groups; groups of exponent 4

Citations:

Zbl 0999.20001