Let p be a prime, P a finite non-cyclic p-gvoup and (xd)Q>(P) of the factor group P/$(P), and the map 61-+B is a homomorphism from the automorphism group Aut P of P to the automorphism group of P/$(P). When P/O(P) is regarded as a vector space over the field of order p, the restric...

For each odd prime p, we produce a 2-generated pro-p group G whose normal Hausdorff spectra hspec ⊴ S ( ) = hdim H ∣ c \begin{equation*}\hskip7pc \operatorname{hspec}_{\trianglelefteq }^{\mathcal {S}}(G) {\left\lbrace \operatorname{hdim}_{G}^{\mathcal {S}}(H)\mid H\trianglelefteq _\mathrm{c} \right\rbrace}\hskip-7pc \end{equation*} with respect to five standard filtration series $\mathcal {S}$ ...

In this paper we consider Q the class of solvable Lie algebras L with the following property: if A is a subalgebra of L, then Φ(A) ⊆ Φ(L) (where Φ(L) denotes the Frattini subalgebra of L;that is Φ(L) is the intersection of all maximal subalgebras of L). The class Q is shown to contain all solvable Lie algebras whose derived algebra is nilpotent. Necessary conditions are found such that an ideal...

In this paper we study the finite dimensional Lie p-algebras, L splitting on its abelian p-socle, the sum of its minimal abelian p-ideals. In addition, some properties of the Frattini p-subalgebra of L are pointed

1. Kristina Altmann, Hyperbolic lines in unitary space. 2. John Bamberg, Transitive m-systems. 3. Barbara Baumeister, The primitive permutation groups with a regular subgroup. 4. Rieuwert Blok, Extensions of isomorphisms for affine grassmannians over F2. 5. Matthew Brown, Tetradic sets of elliptic quadrics of PG(3, q) and generalized quadrangles of order (s, s2) with Property (G). 6. Julia Brow...

Let G $G$ be a finite group and recall that the Frattini subgroup Frat ( ) ${\rm Frat}(G)$ is intersection of all maximal subgroups . In this paper, we investigate number , denoted α $\alpha (G)$ which minimal whose coincides with earlier work, studied in special case where simple here extend analysis to almost groups. particular, prove ⩽ 4 (G) \leqslant 4$ for every best possible. We also esta...

Let L be a Moufang loop which is centrally nilpotent of class 2. We first show that the nuclearly-derived subloop (normal associator subloop) L∗ of L has exponent dividing 6. It follows that Lp (the subloop of L of elements of p-power order) is associative for p > 3. Next, a loop L is said to be a small Frattini Moufang loop, or SFML, if L has a central subgroup Z of order p such that C ∼= L/Z ...

The notions of nearly-maximal and near Frattini subgroups considered by J.B. Riles in [20] and the natural related notions are characterized for abelian groups.