9.76 (1984)

We define a Golod group to be the $r$-generated, $r \geqslant 2$, subgroup $\langle 1 + x_1 + I, 1 + x_2 + I, \dots, 1 + x_r + I \rangle$ of the adjoint group $1 + F/I$ of the factor-algebra $F/I$, where $F$ is a free algebra of polynomials without constant terms in non-commuting variables $x_1, x_2, \dots, x_r$ over a field of characteristic $p > 0$, and $I$ is an ideal of $F$ such that $F/I$ is a non-nilpotent nil-algebra (see E. S. Golod, Amer. Math. Soc. Transl. (2), 48 (1965), 103–106). Prove that in Golod groups the centralizer of every element is infinite.

Note that Golod groups with infinite centre were constructed by A. V. Timofeenko (Math. Notes, 39, no. 5 (1986), 353–355); independently by other methods the same result was obtained in the 90s by L. Hammoudi.