Varieties of metabelian pro-\(p\)-groups (Q1204769)

The main result is an analogue of known results of R. A. Bryce saying that any proper subvariety in the variety of all metabelian pro-\(p\)- groups is either of finite exponent or splits into a union of a variety of finite exponent, a variety of groups with the commutator subgroup of finite exponent \(p^ \alpha\) and a finite union of varieties of groups with nilpotent commutator subgroup and quotient groups of exponent \(p^ \beta\) for appropriate nilpotency class and fixed \(\beta\). A corollary says that the lattice of subvarieties in the variety of all metabelian pro-\(p\)-groups is countable.