In this paper we study G-arc-transitive graphs Δ where the permutation group induced by the stabiliser Gx of the vertex x on the neighbourhood Δ(x) satisfies the two conditions given in the introduction. We show that for such a G-arc-transitive graph Δ, if (x, y) is an arc of Δ, then the subgroup of G fixing Δ(x) and Δ(y) point-wise is a p-group for some prime p. Next we prove that every G-locally primitive (respectively quasiprimitive, semiprimitive) graph satisfies our two local hypotheses. Thus this provides a new Thompson-Wielandt-like theorem for a very large class of arc-transitive graphs.
Furthermore, we give various families of G-arc-transitive graphs where our two local conditions do not apply and where has arbitrarily large composition factors.
A partition for the elements of prime-power order in a finite group G is a family of subgroups with the property that every non-identity element of prime-power order lies in exactly one subgroup of the family. The main result of this paper is a classification of the finite simple groups which have such a partition. We also establish a connection between this concept and the class of permutation groups all of whose elements of prime-power order have the same number of fixed points.
We study the growth of group endomorphisms, a generalization of the classical notion of growth of finitely generated groups, which is strictly related to algebraic entropy. We prove that the inner automorphisms of a group have the same growth type and the same algebraic entropy as the identity automorphism. Moreover, we show that endomorphisms of locally finite groups cannot have intermediate growth. We also find an example showing that the Addition Theorem for algebraic entropy does not hold for endomorphisms of arbitrary groups.
Let G be a finite group, and Irr(G) the set of irreducible complex characters of G. We say that an element g ∈ G is a vanishing element of G if there exists χ in Irr(G) such that χ(g) = 0. In this paper, we consider the set of orders of the vanishing elements of a group G, and we define the prime graph on it, which we denote by Γ(G). Focusing on the class of solvable groups, we prove that Γ(G) has at most two connected components, and we characterize the case when it is disconnected. Moreover, we show that the diameter of Γ(G) is at most 4. Examples are given to round out our understanding of this matter. Among other things, we prove that the bound on the diameter is best possible, and we construct an infinite family of examples showing that there is no universal upper bound on the size of an independent set of Γ(G).
In this paper we are concerned with finite soluble groups G admitting a factorisation , with A and B proper subgroups having coprime order. We are interested in bounding the Fitting height of G in terms of some group-invariants of A and B, including the Fitting heights and the derived lengths.
We show that there exists a constant a such that, for every subgroup H of a finite group G, the number of maximal subgroups of G containing H is bounded above by . In particular, a transitive permutation group of degree n has at most maximal systems of imprimitivity. When G is soluble, generalizing a classic result of Tim Wall, we prove a much stronger bound, that is,
the number of maximal subgroups of G containing H is at most .
We introduce twisted permutation codes, which are frequency permutation arrays
analogous to repetition permutation codes, namely, codes obtained from the repetition construction applied
to a permutation code. In particular, we show that a lower bound for the minimum distance of a twisted permutation code is the minimum distance of a
repetition permutation code. We give examples where this bound is tight, but more importantly, we
give examples of twisted permutation codes with minimum distance strictly greater than this lower bound.