The existence of Golod groups with infinite center
Let p be a prime number, and d be a natural number, \(d\geq 2\). Let F be the associative ring of polynomials without free members in non-commuting variables \(x_ 1,...,x_ d\) over the field GF(p). Let \(f_ 1\), \(f_ 2,..\). be homogeneous polynomials in F, I be the ideal generated by these elements, and \(r_ i\) be the number of polynomials of degree i in the sequence \(f_ 1\), \(f_ 2,..\). If \(r_ i\leq \epsilon^ 2(d- 2\epsilon)^{i-2}\), \(0<\epsilon <0,5\), and F/I is a nil-algebra, then the multiplicative group \(G_ I=gp(1+x_ 1+I,...,1+x_ d+I)\) is just the Golod group studied in this article where 1 is the unity of the field GF(p). It is proved that, for every prime number p and every natural number \(d\geq 2\), there exists a d-generated Golod p-group \(G_ I\) all of whose (d-1)-generated subgroups are finite such that its center \(Z(G_ I)\) is infinite.
scientific article; zbMATH DE number 3973160