Lie nilpotency indices of modular group algebras

We give an explicit list of all p-groups G with a cyclic subgroup of index p^2, such that the group algebra KG over the field K of characteristic p has a filtered multiplicative K-basis. We also proved that such a K-basis does not exist for the group algebra KG, in the case when G$ is either a powerful p-group or a two generated p-group (p\not=2) with a central cyclic commutator subgroup. This paper is a continuation of the related V. Bovdi, On a filtered multiplicative basis...

Let $G$ be a finite group. If $M_n < M_{n-1} < \ldots < M_1 < M_{0}=G $ where $M_i$ is a maximal subgroup of $M_{i-1}$ for all $i=1, \ldots ,n$, then $M_n $ ($n > 0$) is an \emph{$n$-maximal subgroup} of $G$. A subgroup $M$ of $G$ is called \emph{modular} if the following conditions are held: (i) $\langle X, M \cap Z \rangle=\langle X, M \rangle \cap Z$ for all $X \leq G, Z \leq G$ such that $X \leq Z$, and (ii) $\langle M, Y \cap Z \rangle=\langle M, Y \rangle \cap Z$ fo...

In this paper we describe an algorithm for finding the nilpotency class, and the upper central series of the maximal normal p-subgroup N(G) of the automorphism group, Aut(G) of a bounded (or finite) abelian p-group G. This is the first part of two papers devoted to compute the nilpotency class of N(G) using formulas, and algorithms that work in almost all groups. Here, we prove that for p>2 the algorithm always runs. The algorithm describes a sequence of ideals of the Jacobso...

We study the Modular Isomorphism Problem applying a combination of existing and new techniques. We make use of the small group algebra to give a positive answer for two classes of groups of nilpotency class 3. We also introduce a new approach to derive properties of the lower central series of a finite $p$-group from the structure of the associated modular group algebra. Finally, we study the class of so-called $p$-obelisks which are highlighted by recent computer-aided inves...

Let $G$ be a group and let $F$ be a field of characteristic different from 2. Denote by $(FG)^+$ the set of symmetric elements and by $\mathcal{U}^+(FG)$ the set of symmetric units, under an oriented classical involution of the group algebra $FG$. We give some lower and upper bounds on the Lie nilpotency index of $(FG)^+$ and the nilpotency class of $\mathcal{U}^+(FG)$.

A classification of finite groups in which every 3-maximal subgroup is K-U-subnormal is given.

We adopt the $p$-group generation algorithm to classify small-dimensional nilpotent Lie algebras over small fields. Using an implementation of this algorithm, we list the nilpotent Lie algebras of dimension at most~9 over $\F_2$ and those of dimension at most~7 over $\F_3$ and $\F_5$.

We prove that if $G$ is finite 2-generated $p$-group of nilpotence class at most 2 then the group algebra of $G$ with coefficients in the field with $p$ elements determines $G$ up to isomorphisms.

Let $n$ be a natural number. A subgroup $H$ of a group $G$ will be called $n$-modularly embedded in $G$ if either $H$ is normal in $G$ or $H\not= \mathrm{Core}_{G}(H)$, $|G:H| =p$ and $|G/\mathrm{Core}_{G}(H)|=pq^{n}$, $q^{n}$ divides $p-1$ for some primes $p$ and $q$. Let $k$ be a fixed natural number. A subgroup $H$ of a group $G$ will be called $k$-submodular in $G$ if there exists a chain $H = H_{0} \leq H_{1} \leq \cdots \leq H_{m-1} \leq H_{m} = G$ of subgroups such tha...

We give an explicit list of all p-groups G of order at most p^4 or 2^5 such that the group algebra KG over the field K of characteristic p has a filtered multiplicative K-basis.