A group is called capable if it is a central factor group. We consider the capability of nilpotent products of cyclic groups, and obtain a generalisation of a theorem of Baer for the small class case. The approach may also be used to obtain some recent results on the capability of certain nilpotent groups of class 2. We also obtain a necessary condition for the capability of an arbitrary p-grou...

We announce proofs of a number of theorems concerning finite p-groups and nilpotent groups. These include: (1) the number of p-groups of class c on d generators of order pn satisfies a linear recurrence relation in n; (2) for fixed n the number of p-groups of order pn as one varies p is given by counting points on certain varieties mod p; (3) an asymptotic formula for the number of finite nilpo...

By the Shepherd–Leedham-Green–McKay theorem on finite p-groups of maximal class, if a finite p-group of order pn has nilpotency class n−1, then it has a subgroup of nilpotency class at most 2 with index bounded in terms of p. Counterexamples to a rank analogue of this theorem are constructed, which give a negative solution to Problem 16.103 in Kourovka Notebook. Moreover, it is shown that there...

Let KG be the group ring of a group G over a field of characteristic p > 0, p ^ 2, 3. Suppose G contains no element of order p (if p > 0). Group algebras KG with unit group U(KG) solvable and n-Engel are characterized. Let ATG be the group ring of a group G over a field K of characteristic p > 0 and let U(KG) denote its group of units. Several authors including Bateman [1], Bateman and Coleman ...

In this paper, we prove the conjectures made in a joint paper of the author with Carlson and Robinson, on the vanishing of cohomology of a nite group G. In particular, we prove that if k is a eld of characteristic p, then every non-projective kG-module M in the principal block has nontrivial cohomology in the sense that H (G;M) 6= 0, if and only if the centralizer in G of every element of order...

Let G be a connected linear semisimple Lie group with Lie algebra g, and let K C → Aut(p C ) be the complexified isotropy representation at the identity coset of the corresponding symmetric space. Suppose that Ω is a nilpotent G-orbit in g and O is the nilpotent K C -orbit in p C associated to Ω by the Kostant-Sekiguchi correspondence. We show that the complexity of O as a K C variety measures ...

Let G be a connected reductive linear algebraic group defined over an algebraically closed field of characteristic p. Assume that p is good for G. In this note we classify all the spherical nilpotent G-orbits in the Lie algebra of G. The classification is the same as in the characteristic zero case obtained by D.I. Panyushev in 1994, [32]: for e a nilpotent element in the Lie algebra of G, the ...

We prove that the centralizer Cφ ⊆ HomR(M,M) of a nilpotent endomorphism φ of a finitely generated semisimple left R-module RM (over an arbitrary ring R) is the homomorphic image of the opposite of a certain Z(R)-subalgebra of the full m×m matrix algebra Mm×m(R[t]), where m is the dimension (composition length) of ker(φ). If R is a finite dimensional division ring over its central subfield Z(R)...

In this paper, we prove the conjectures made in a joint paper of the author with Carlson and Robinson, on the vanishing of cohomology of a nite group G. In particular, we prove that if k is a eld of characteristic p, then every non-projective kG-module M in the principal block has nontrivial cohomology in the sense that H (G; M) 6 = 0, if and only if the centralizer in G of every element of ord...

Let G be an elementary abelian p-group of rank n, with p an odd prime. In order to count the Hopf Galois structures of type G on a Galois extension of fields with Galois group G, we need to determine the orbits under conjugation by Aut(G) of regular subgroups of the holomorph of G that are isomorphic to G. The orbits correspond to isomorphism types of commutative nilpotent Fp-algebras N of dime...