In this paper, we define a generalized Wielandt subgroup, local generalized Wielandt subgroup and its series for finite group and discuss its different basic properties which explain the notion of generalized Wielandt subgroup in a better way. We bound generalized Wielandt length as a function of nilpotency classes of its Sylow subgroups.

We study Kuzmin’s conjecture on the index of nilpotency for the variety N il5 of associative nil-algebras of degree 5. Due to Vaughan-Lee [11] the problem is reduced to that for k-generator N il5-superalgebras, where k ≤ 5. We confirm Kuzmin’s conjecture for 2-generator superalgebras proving that they are nilpotent of degree 15.

Questions about nilpotency of groups satisfying Engel conditions have been considered since 1936, when Zorn proved that finite Engel groups are nilpotent. We prove that 4-Engel groups are locally nilpotent. Our proof makes substantial use of both hand and machine calculations.

The descent algebra of the symmetric group, over a field of non-zero characteristic p, is studied. A homomorphism into the algebra of generalised p-modular characters of the symmetric group is defined. This is then used to determine the radical, and its nilpotency index. It also allows the irreducible representations of the descent algebra to be described.

We consider the supersymmetric WZNW model gauged in a manifestly supersymmetric way. We find the BRST charge and the necessary condition for nilpotency. In the BRST framework the model proves to be a Lagrangian formulation of the supersymmetric coset construction, known as the N=1 Kazama-Suzuki coset construction.