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13.38 (1995)

Solved

Let $G = F/R$ be a pro-$p$-group with one defining relation, where $R$ is the normal subgroup of a free pro-$p$-group $F$ generated by a single element $r \in F^p[F, F]$.
$\qquad$ a) Suppose that $r = t^p$ for some $t \in F$; can $G$ contain a Demushkin group as a subgroup?
$\qquad$ b) Do there exist two pro-$p$-groups $G_1 \supset G_2$ with one defining relation, where $G_1$ has elements of finite order, while the subgroup $G_2$ is torsion-free?

Contributor: O. V. Mel’nikov

Progress

a) Yes, it can. b) Yes, they exist. The pro-$p$-group with presentation $\langle a, b \mid [a, b]^p = 1 \rangle$ contains the Demushkin group $\langle x_1, \dots, x_{2g} \mid \prod_{i=1}^g [x_{2i-1}, x_{2i}] = 1 \rangle$ for some $g > 1$. (O. V. Mel’nikov, Letter of February, 28, 1999.)

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