## 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

Zbl 0999.20001
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