MGMichael Giudici

Michael Giudici

The University of Western Australia
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Michael Giudici has a total of 137 co-authors. Here are the top co-authors by number of shared publications:

Latest Publications

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    Prime power coverings of groups and field extensions

    by Michael Giudici, Luke Morgan, Cheryl E. Praeger

    For a finite group A with normal subgroup G , a subgroup U of G is an A -prime-power-covering subgroup if U meets every A -conjugacy-class of elements of G of prime-power order. It is conjectured that | G : U | $|G:U|$ StartAbsoluteValue upper G colon upper U EndAbsoluteValue is bounded by some function of | A : G | $|A:G|$ StartAbsoluteValue upper A colon upper G EndAbsoluteValue . We prove the conjecture in the case that the action of G on the set of right cosets of U in G is innately transitive. This includes the special case where U is a maximal subgroup of G . It turns out that each prime-power-covering subgroup U for which this G -action is innately transitive must be a maximal subgroup of G . We derive the following number-theoretic consequence: if K is a finite Galois extension of a global field k and L is a finite innate extension of K such that K and L are weakly Kronecker equivalent over k , then the extension L / K $L/K$ upper L divided by upper K must be atomic of degree bounded in terms of [ K : k ] $[K:k]$ left bracket upper K colon k right bracket . Our proofs use both a recently proved bound on the order of a finite nonabelian simple group in terms of the number of its classes of elements of prime-power order (depending on the Finite Simple Group Classification) and also new Galois-theoretic concepts for field extensions.

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    Linear dimension of group actions

    by Alice Devillers, Michael Giudici, Daniel R. Hawtin, Lukas Klawuhn, Luke Morgan

    Two fundamental ways to represent a group are as permutations and as matrices. In this paper, we study linear representations of groups that intertwine with a permutation representation. Recently, D'Alconzo and Di Scala investigated how small the matrices in such a linear representation can be. The minimal dimension of such a representation is the \emph{linear dimension of the group action} and this has applications in cryptography and cryptosystems. We develop the idea of linear dimension from an algebraic point of view by using the theory of permutation modules. We give structural results about representations of minimal dimension and investigate the implications of faithfulness, transitivity and primitivity on the linear dimension. Furthermore, we compute the linear dimension of several classes of finite primitive permutation groups. We also study wreath products, allowing us to determine the linear dimension of imprimitive group actions. Finally, we give the linear dimension of almost simple finite $2$-transitive groups, some of which may be used for further applications in cryptography. Our results also open up many new questions about linear representations of group actions.

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    Finite simple groups have many classes of p-elements

    by Michael Giudici, Luke Morgan, Cheryl E. Praeger

    For an element x of a finite group T , the Aut(T)-class of x is {x σ | σ ∈ Aut(T)}. We prove that the order |T | of a finite nonabelian simple group T is bounded above by a function of the parameter m(T), where m(T) is the maximum, over all primes p, of the number of Aut(T)-classes of elements of T of p-power order. This bound is a substantial generalisation of the results of Pyber (1992) and of Héthelyi and Külshammer (2005), and it has implications for relative Brauer groups of finite extensions of global fields.

    Cited by 4
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    Low Rank Groups of Lie Type Acting Point- and Line-Primitively on Finite Generalised Quadrangles

    by Vishnuram Arumugam, John Bamberg, Michael Giudici

    Suppose we have a finite thick generalised quadrangle whose automorphism group $G$ acts primitively on both the set of points and the set of lines. Then $G$ must be almost simple. In this paper, we show that $\operatorname{soc}(G)$ cannot be isomorphic to $\operatorname{Sz}(2^{2m+1})$ or $\operatorname{Ree}(3^{2m+1})$ where $m$ is a positive integer.

    Cited by 1
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    Bounding s for vertex-primitive s-arc-transitive digraphs of alternating and symmetric groups

    by Junyan Chen, Lei Chen, Michael Giudici, Jingjian Li, Cheryl E. Praeger, Binzhou Xia

    AMC is a journal published by University of Primorska.

    Cited by 3
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    Tactical decompositions in finite polar spaces and non-spreading classical group actions

    by John Bamberg, Michael Giudici, Jesse Lansdown, Gordon F. Royle

    For finite classical groups acting naturally on the set of points of their ambient polar spaces, the symmetry properties of synchronising and separating are equivalent to natural and well-studied problems on the existence of certain configurations in finite geometry. The more general class of spreading permutation groups is harder to describe, and it is the purpose of this paper to explore this property for finite classical groups. In particular, we show that for most finite classical groups, their natural action on the points of its polar space is non-spreading. We develop and use a result on tactical decompositions (an AB-Lemma) that provides a useful technique for finding witnesses for non-spreading permutation groups. We also consider some of the other primitive actions of the classical groups.

    Cited by 5
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    Vertex-primitive s-arc-transitive digraphs of symplectic groups

    by Lei Chen, Michael Giudici, Cheryl E. Praeger

    A digraph is $s$-arc-transitive if its automorphism group is transitive on directed paths with $s$ edges, that is, on $s$-arcs. Although infinite families of finite $s$-arc transitive digraphs of arbitrary valency were constructed by the third author in 1989, existence of a vertex-primitive $2$-arc-transitive digraph was not known until an infinite family was constructed by the second author with Li and Xia in 2017. This led to a conjecture by the second author and Xia in 2018 that, for a finite vertex-primitive $s$-arc-transitive digraph, $s$ is at most $2$, together with their proof that it is sufficient to prove the conjecture for digraphs with an almost simple group of automorphisms. This paper confirms the conjecture for finite symplectic groups.

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    Spreading primitive groups of diagonal type do not exist

    by John Bamberg, Saul D. Freedman, Michael Giudici

    The synchronization hierarchy of finite permutation groups consists of classes of groups lying between $2$ -transitive groups and primitive groups. This includes the class of spreading groups, which are defined in terms of sets and multisets of permuted points, and which are known to be primitive of almost simple, affine or diagonal type. In this paper, we prove that in fact no spreading group of diagonal type exists. As part of our proof, we show that all non-abelian finite simple groups, other than six sporadic groups, have a transitive action in which a proper normal subgroup of a point stabilizer is supplemented by all corresponding two-point stabilizers.

    Cited by 4
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    Subgroups of classical groups that are transitive on subspaces

    by Michael Giudici, S.P. Glasby, Cheryl E. Praeger

    Cited by 19
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    A Generalization of Szep’s Conjecture for Almost Simple Groups

    by Nick Gill, Michael Giudici, Pablo Spiga

    We prove a natural generalization of Szep’s conjecture. Given an almost simple group G with socle not isomorphic to an orthogonal group having Witt defect zero, we classify all possible group elements $$x,y\in G\setminus \{1\}$$ x , y ∈ G \ { 1 } with $$G=\textbf{N}_G(\langle x\rangle )\textbf{N}_G(\langle y\rangle )$$ G = N G ( ⟨ x ⟩ ) N G ( ⟨ y ⟩ ) , where we are denoting by $$\textbf{N}_G(\langle x\rangle )$$ N G ( ⟨ x ⟩ ) and by $$\textbf{N}_G(\langle y\rangle )$$ N G ( ⟨ y ⟩ ) the normalizers of the cyclic subgroups $$\langle x\rangle $$ ⟨ x ⟩ and $$\langle y\rangle $$ ⟨ y ⟩ . As a consequence of this result, we classify all possible group elements $$x,y\in G\setminus \{1\}$$ x , y ∈ G \ { 1 } with $$G=\textbf{C}_G(x)\textbf{C}_G(y)$$ G = C G ( x ) C G ( y ) .

    Cited by 4
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