ON THE RANK OF A VERBAL SUBGROUP OF A FINITE GROUP

We show that if w is a multilinear commutator word and G a finite group in which every metanilpotent subgroup generated by w-values is of rank at most r, then the rank of the verbal subgroup $w(G)$ is bounded in terms of r and w only. In the case where G is soluble, we obtain a better result: if G is a finite soluble group in which every nilpotent subgroup generated by w-values is of rank at most r, then the rank of $w(G)$ is at most $r+1$ .

The Projective Character Tables of a Solvable Group 26:6×2

The Chevalley–Dickson simple group G24 of Lie type G2 over the Galois field GF4 and of order 251596800=212.33.52.7.13 has a class of maximal subgroups of the form 24+6:A5×3, where 24+6 is a special 2-group with center Z24+6=24. Since 24 is normal in 24+6:A5×3, the group 24+6:A5×3 can be constructed as a nonsplit extension group of the form G¯=24·26:A5×3. Two inertia factor groups, H1=26:A5×3 and H2=26:6×2, are obtained if G¯ acts on 24. In this paper, the author presents a method to compute all projective character tables of H2. These tables become very useful if one wants to construct the ordinary character table of G¯ by means of Fischer–Clifford theory. The method presented here is very effective to compute the irreducible projective character tables of a finite soluble group of manageable size.

ON A LATTICE CHARACTERISATION OF FINITE SOLUBLE PST-GROUPS

Let $\mathfrak{F}$ be a class of finite groups and $G$ a finite group. Let ${\mathcal{L}}_{\mathfrak{F}}(G)$ be the set of all subgroups $A$ of $G$ with $A^{G}/A_{G}\in \mathfrak{F}$. A chief factor $H/K$ of $G$ is $\mathfrak{F}$-central in $G$ if $(H/K)\rtimes (G/C_{G}(H/K))\in \mathfrak{F}$. We study the structure of $G$ under the hypothesis that every chief factor of $G$ between $A_{G}$ and $A^{G}$ is $\mathfrak{F}$-central in $G$ for every subgroup $A\in {\mathcal{L}}_{\mathfrak{F}}(G)$. As an application, we prove that a finite soluble group $G$ is a PST-group if and only if $A^{G}/A_{G}\leq Z_{\infty }(G/A_{G})$ for every subgroup $A\in {\mathcal{L}}_{\mathfrak{N}}(G)$, where $\mathfrak{N}$ is the class of all nilpotent groups.

On cofactors of subnormal subgroups

For a soluble finite group [Formula: see text] and a prime [Formula: see text] we let [Formula: see text], [Formula: see text]. We obtain upper bounds for the rank, the nilpotent length, the derived length, and the [Formula: see text]-length of a finite soluble group [Formula: see text] in terms of [Formula: see text] and [Formula: see text].