We address the question of the dualizability of nilpotent Mal’cev algebras, showing that nilpotent finite Mal’cev algebras with a nonabelian supernilpotent congruence are inherently nondualizable. In particular, finite nilpotent nonabelian Mal’cev algebras of finite type are nondualizable if they are direct products of algebras of prime power order. We show that these results cannot be generalized to nilpotent algebras by giving an example of a group expansion of infinite type that is nilpotent and nonabelian, but dualizable. To our knowledge this is the first construction of a nonabelian nilpotent dualizable algebra. It has the curious property that all its nonabelian finitary reducts with group operation are nondualizable. We were able to prove dualizability by utilizing a new clone theoretic approach developed by Davey, Pitkethly, and Willard. Our results suggest that supernilpotence plays an important role in characterizing dualizability among Mal’cev algebras.

The purpose of this paper is to study the influence of $c$-supplemented minimal subgroups on the $p$-nilpotency of finite groups. We obtain ‘iff' and ‘localized' versions of theorems of Itô and Buckley on nilpotence, $p$-nilpotence and supersolvability.

K. Ding studied a class of Schubert varieties ${{X}_{\lambda }}$ in type A partial flag manifolds, indexed by integer partitions $\text{ }\!\!\lambda\!\!\text{ }$ and in bijection with dominant permutations. He observed that the Schubert cell structure of ${{X}_{\lambda }}$ is indexed by maximal rook placements on the Ferrers board ${{B}_{\lambda \text{ }}}$, and that the integral cohomology groups ${{H}^{*}}\left( {{X}_{\lambda }};\,\mathbb{Z} \right),\,{{H}^{*}}\left( {{X}_{\mu }};\,\mathbb{Z} \right)$ are additively isomorphic exactly when the Ferrers boards ${{B}_{\lambda \text{ }}}$, ${{B}_{\mu }}$ satisfy the combinatorial condition of rook-equivalence.

We classify the varieties ${{X}_{\lambda }}$ up to isomorphism, distinguishing them by their graded cohomology rings with integer coefficients. The crux of our approach is studying the nilpotence orders of linear forms in the cohomology ring.