In this paper we continue developing the theory of symplectic alternating algebras that was started in [3]. We focus on nilpotency, solubility and nil-algebras. We show in particular that symplectic alternating nil-2 algebras are always nilpotent and classify all nil-algebras of dimension up to 8.

Felix and Murillo [5] introduced the group AutΩ(X) of self-maps f of X, which satisfy Ωf = 1ΩX , and proved that the group is nilpotent with the order of nilpotency bounded by the Lusternik-Schnirelmann category of X. In this paper we construct a spectral sequence converging to the group AutΩ(X) and derive several interesting consequences.

We consider Mn(R), the ring of n by n matrices on the real numbers where n ≥ 2. We find all of the natural numbers that are the degree of nilpotency of some nilpotent derivations on Mn(R). Mathematics Subject Classification: 16S50, 13N15, 16N40

We prove analogues of results of Tate and Yoshida on control of transfer for fusion systems. This requires the notions of p-group residuals and transfer maps in cohomology for fusion systems. As a corollary we obtain a p-nilpotency criterion due to Tate.

We consider the complexity of computing the determinant over arbitrary finite-dimensional algebras. We first consider the case that A is fixed. We obtain the following dichotomy: If A/ radA is noncommutative, then computing the determinant over A is hard. “Hard” here means #P-hard over fields of characteristic 0 and ModpP-hard over fields of characteristic p > 0. If A/ radA is commutative and t...

Abstract In this article, some new sufficient conditions of p -nilpotency finite groups are obtained by using c -normality and Φ-supplementary the maximal or 2-maximal subgroups Sylow -subgroups.

Abstract We define isoclinism of skew braces and present several applications. study some properties that are invariant under isoclinism. For example, we prove right nilpotency is an invariant. This result has application in the theory set‐theoretic solutions to Yang–Baxter equation. isoclinic multipermutation