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

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 \(k=3\).

MSC:

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

Citations:

Zbl 0999.20001
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