A pseudo-hyperoval of a projective space PG(3n − 1, q), q even, is a set of q n + 2 subspaces of dimension n − 1 such that any three span the whole space. We prove that a pseudohyperoval with an irreducible transitive stabilizer is elementary. We then deduce from this result a classification of the thick generalized quadrangles Q that admit a point-primitive, linetransitive automorphism group with a point-regular abelian normal subgroup. Specifically, we show that Q is flag-transitive and isomorphic to T * 2 (H), where H is either the regular hyperoval of PG(2, 4) or the Lunelli-Sce hyperoval of PG(2, 16). (i) H acts transitively on the lines of Q;(ii) H acts transitively on the flags of Q; (iii) H acts transitively on the pseudo-hyperoval {U 1 , U 2 , . . . , U t+1 }, by conjugation.Theorem 1.1 is proved in Section 2. It implies that a classification of transitive pseudohyperovals would yield a classification of the generalized quadrangles that admit a line-transitive automorphism group with a point-regular abelian normal subgroup, and, moreover, that such generalized quadrangles are, in fact, flag-transitive. By a result of J. A. Thas [28, §4.5], if a projective space PG(3n − 1, q) contains a pseudo-hyperoval, then q = 2 f for some positive integer f . For small values of the product nf , we appeal to some existing results to classify the transitive pseudo-hyperovals of PG(3n − 1, 2 f ), Journal of Combinatorial Designs

Let G = X be an absolutely irreducible subgroup of GL(d, K), and let F be a proper subfield of the finite field K. We present a practical algorithm to decide constructively whether or not G is conjugate to a subgroup of GL(d, F ).K × , where K × denotes the centre of GL(d, K). If the derived group of G also acts absolutely irreducibly, then the algorithm is Las Vegas and costs O(|X|d 3 + d 2 log |F |) arithmetic operations in K. This work forms part of a recognition project based on Aschbacher's classification of maximal subgroups of GL(d, K).

The chief aim of this paper is to describe a procedure which, given a d-dimensional absolutely irreducible matrix representation of a finite group over a finite field E, produces an equivalent representation such that all matrix entries lie in a subfield F of E which is as small as possible. The algorithm relies on a matrix version of Hilbert's Theorem 90, and is probabilistic with expected running time O(|E : F|d 3 ) when |F| is bounded. Using similar methods we then describe an algorithm which takes as input a prime number and a power-conjugate presentation for a finite soluble group, and as output produces a full set of absolutely irreducible representations of the group over fields whose characteristic is the specified prime, each representation being written over its minimal field.

The only known skew-translation generalised quadrangles (STGQ) having order (q, q), with q even, are translation generalised quadrangles. Equivalently, the only known groups G of order q 3 , q even, admitting an Ahrens-Szekeres (AS-)configuration are elementary abelian. In this paper we prove results in the theory of STGQ giving (i) new structural information for a group G admitting an AS-configuration, (ii) a classification of the STGQ of order (8,8), and (iii) a classification of the STGQ of order (q, q) for odd q (using work of Ghinelli and Yoshiara).Crown

Given a field F , a scalar λ ∈ F and a matrix A ∈ F n×n , the twisted central-where c A (t) denotes the characteristic polynomial of A. We also show how C F (A, λ) decomposes, and we estimate the probability that C F (A, λ) is nonzero when |F | is finite. Finally, we prove dim C F (A, λ) n 2 /2 for λ ∈ {0, 1} and 'almost all' n × n matrices A over F .

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This paper investigates alternation patterns in length, shape and orientation of dorsal cirri (fleshy segmental appendages) of phyllodocidans, a large group of polychaete worms (Annelida). We document the alternation patterns in several families of Phyllodocida (Syllidae, Hesionidae, Sigalionidae, Polynoidae, Aphroditidae and Acoetidae) and identify the simple mathematical rule bases that describe the progression of these sequences. Two fundamentally different binary alternation patterns were found on the first four segments: 1011 for nereidiform families and 1010 for aphroditiform families. The alternation pattern in all aphroditiform families matches a simple one-dimensional cellular automaton and that for Syllidae (nereidiform) matches the Fibonacci string sequence. Hesionidae (nereidiform) showed the greatest variation in alternation patterns, but all corresponded to various known substitution rules. Comparison of binary patterns of the first 22 segments using a distance measure supports the current ideas on phylogeny within Phyllodocida. These results suggest that gene(s) involved in postlarval segmental growth employ a switching sequence that corresponds to simple mathematical substitution rules.

We consider the structure of finite p-groups G having precisely three characteristic subgroups, namely 1, Φ(G) and G. The structure of G varies markedly depending on whether G has exponent p or p 2 , and, in both cases, the study of such groups raises deep problems in representation theory. We present classification theorems for 3-and 4-generator groups, and we also study the existence of such r-generator groups with exponent p 2 for various values of r. The automorphism group induced on the Frattini quotient is, in various cases, related to a maximal linear group in Aschbacher's classification scheme.