Constructing normalisers in finite soluble groups

AbstractThis paper describes algorithms for constructing a Hall π-subgroup H of a finite soluble group G and the normaliser NG(H). If G has composition length n, then H and NG(H) can be constructed using O(n4 log |G|) and O(n5 log |G|) group multiplications, respectively. These algorithms may be used to construct other important subgroups such as Carter subgroups, system normalisers and relative system normalisers. Computer implementations of these algorithms can compute a Sylow 3-subgroup of a group with n = 84, and its normaliser in 47 seconds and 30 seconds, respectively. Constructing normalisers of arbitrary subgroups of a finite soluble group can be complicated. This is shown by an example where constructing a normaliser is equivalent to constructing a discrete logarithm in a finite field. However, there are no known polynomial algorithms for constructing discrete logarithms

Point-primitive generalised hexagons and octagons

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Conjugacy class sizes in arithmetic progression

Let cs(G) denote the set of conjugacy class sizes of a group G, and let cs 17 (G) = cs(G){1} be the sizes of non-central classes. We prove three results. We classify all finite groups for which (1) cs (G) = {a, a + d, ef, a + r d} is an arithmetic progression with r 65 2 (2) cs 17 (G) = { 2, 4, 6 } (G)= {2,4,6} is the smallest case where cs 17 (G) is an arithmetic progression of length more than 2 (our most substantial result); (3) the largest two members of cs 17 (G) are coprime. For (3), it is not obvious, but it is true that cs 17(G) has two elements, and so is an arithmetic progression. \ua9 2020 Walter de Gruyter GmbH, Berlin/Boston 2020 Australian Research Council DP190100450 Engineering and Physical Sciences Research Council EP/R014604/1 SG and CP gratefully acknowledge support from the Australian Research Council Discovery Project DP190100450. MB acknowledges support from G.N.S.A.G.A. (Indam) and thanks the Centre for the Mathematics of Symmetry and Computation (CMSC) for its hospitality. This work began in the CMSC Research Retreat of 2019. CP also thanks the Isaac Newton Institute for Mathematical Sciences for support and hospitality during the programme "Groups, Representations and applications: New Perspectives". This program was supported by EPSRC grant number EP/R014604/1

Using recurrence relations to count certain elements in symmetric groups

We use the fact that certain cosets of the stabilizer of points are pairwise conjugate in a symmetric group Sn in order to construct recurrence relations for enumerating certain subsets of Sn. Occasionally one can find ‘closed form’ solutions to such recurrence relations. For example, the probability that a random element of Sn has no cycle of length divisible by q is ∏d = 1⌊n / q⌋(1 − 1dq)

Towards an efficientMeat-axealgorithm usingf-cyclicmatrices: The density of uncyclic matrices in M(n,q)

An element X in the algebra M(n,F) of all n × n matrices over a field F is said to be f-cyclic if the underlying vector space considered as an F[X]-module has at least one cyclic primary component. These are the matrices considered to be “good” in the Holt–Rees version of Norton’s irreducibility test in the Meat-axe algorithm. We prove that, for any finite field Fq, the proportion of matrices in M (n,Fq)that are “not good” decays exponentially to zero as the dimension n approaches infinity. Turning this around, we prove that the density of “good” matrices in M(n,Fq) for the Meat-axe depends on the degree, showing that it is at least 1− (2/q) ((1/q)+(1/q2)+(2/q3))n for q ≥ 4. We conjecture that the density is at least 1−(1/q)((1/q)+(1/2q2))n for all q and n, and confirm this conjecture for dimensions n ≤ 37. Finally we give a one-sided Monte Carlo algorithm called Is f Cyclic to test whether a matrix is “good,” at a cost of O(Mat(n) log n) field operations, where Mat(n) is an upper bound for the number of field operations required to multiply two matrices in M(n,Fq)

p-Groups with a unique proper non-trivial characteristic subgroup

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 p2, 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 p2 for various values of r. The automorphism group induced on the Frattini quotient is, in various cases, related to a maximal linear group in Aschbacherʼs classification scheme