11.10 (1990)
Partially Solved(R. C. Lyndon).
a) Does there exist an algorithm that, given a group word $w(a, x)$, recognizes whether $a$ is equal to the identity element in the group $\langle a, x \mid a^n = 1, \ w(a, x) = 1 \rangle$?
b) Is it true that $a \neq 1$ in $G = \langle a, x \mid a^5 = 1, a^{x^2} = [a, a^x] \rangle$?
Progress
b) Yes, it is true, since the mapping $a \to (1 3 5 2 4), x \to (1 2 4 3 5)$ can be extended to a homomorphism of the group $G$ onto the alternating group $A_5$ (D. N. Azarov, in: Algebraicheskije Systemy, Ivanovo, 1991, 4–5 (Russian)).
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