Related papers: On 2-groups of almost maximal class

We give an explicit description of nilpotent Chernikov 2-groups with elementary tops and the basis of rank 2.

It is shown that a finite group in which more than 3/4 of the elements are involutions must be an elementary abelian 2-group. A group in which exactly 3/4 of the elements are involutions is characterized as the direct product of the…

We consider the capability of $p$-groups of class two and odd prime exponent. The question of capability is shown to be equivalent to a statement about vector spaces and linear transformations, and using the equivalence we give proofs of…

We classify Nichols algebras of irreducible Yetter-Drinfeld modules over nonabelian groups satisfying an inequality for the dimension of the homogeneous subspace of degree two. All such Nichols algebras are finite-dimensional, and all known…

Let $G$ be an odd order nilpotent group with class 2 and $e$ denotes the exponent of its commutator subgroup. Let $e=p_1^{r_1}p_2^{r_2}... p_s^{r_s}$, where $p_i$'s are odd primes and $r_i$'s are non-negative integers. Then there are at…

Over an algebraically closed base field $k$ of characteristic 2, the ring $R^G$ of invariants is studied, $G$ being the orthogonal group O(n) or the special orthogonal group SO(n) and acting naturally on the coordinate ring $R$ of the…

In this paper, for n a positve integer, we compute the number of n degree representations for a dihedral group G of order 2m, m \geq 3 and the dimensions of the corresponding spaces of G invariant bilinear forms over a complex field C. We…

The main theorem in this article shows that a group of odd order which admits the alternating group of degree 5 with an element of order 5 acting fixed point freely is nilpotent of class at most two. For all odd primes r, other than 5, we…

Let G be a group of order 8 and F an algebraically closed field with char= 2. In this paper we compute the number of n degree representations of G and subsequent dimensions of the corresponding spaces of invatiant bilinear forms over the…

We describe maximal nilpotent subsemigroups of a given nilpotency class in the semigroup $\Omega_n$ of all $n\times n$ real matrices with non-negative coefficients and the semigroup $\mathbf{D}_n$ of all doubly stochastic real matrices.

A (2,*)-group is a group that can be generated by two elements, one of which is an involution. We describe the method we have used to produce a census of all (2,*)-groups of order at most 6 000. Various well-known combinatorial structures…

In the present paper we study the rigidity of 2-step Carnot groups, or equivalently, of graded 2-step nilpotent Lie algebras. We prove the alternative that depending on bi-dimensions of the algebra, the Lie algebra structure makes it either…

Let K be a field of positive characteristic p, let R be either a group algebra K[G] or a restricted enveloping algebra u(L), and let I be the augmentation ideal of R. We first characterize those R for which I satisfies a polynomial identity…

The invariant subalgebra H^+ of the Heisenberg vertex algebra H under its automorphism group Z/2Z was shown by Dong-Nagatomo to be a W-algebra of type W(2,4). Similarly, the rank n Heisenberg vertex algebra H(n) has the orthogonal group…

We study those $(2,m,n)$-groups which are almost simple and for which the absolute value of the Euler characteristic is a product of two prime powers. All such groups which are not isomorphic to $PSL_2(q)$ or $PGL_2(q)$ are completely…

The exact degree bound for the generators of rings of polynomial invariants is determined for the finite, non-cyclic groups having a cyclic subgroup of index two. It is proved that the Noether number of these groups equals one half the…

In this paper we classify the isomorphism classes of four dimensional nilpotent associative algebras over a field F, studying regular subgroups of the affine group AGL_4(F). In particular we provide explicit representatives for such classes…

We show that every finite group $G$ of size at least $3$ has a nilpotent subgroup of class at most $2$ and size at least $|G|^{1/32\log\log|G|}$. This answers a question of Pyber, and is essentially best possible.

In this paper, we classify (n+5)-dimensional nilpotent n-Lie algebras of class two over the arbitrary field, when $n\ge 3$.

Given a finite group $G$, we denote by $\nu(G)$ the probability that two randomly chosen elements of $G$ generate a nilpotent subgroup. We prove that if $\nu(G)>1/12,$ then $G$ is solvable.