Critical groups having central monoliths of a nilpotent by abelian product variety of groups

Abstract:

Let $\mathfrak {N}$ be a variety of groups which has nilpotency class two and finite odd exponent. Let $\mathfrak {A}$ be an abelian variety of groups with finite exponent relatively prime to the exponent of $\mathfrak {N}$. The existence in the product variety $\mathfrak {N}\mathfrak {A}$ of nonnilpotent critical groups having central monoliths is established. The structure of these critical groups is studied. This structure is shown to depend on an invariant, k. The join-irreducible subvariety of $\mathfrak {N}\mathfrak {A}$ generated by the nonnilpotent critical groups of $\mathfrak {N}\mathfrak {A}$ having central monoliths is determined, in particular, for k odd.