Let G be a nontrivial transitive permutation group on a finite set Ω. An element of G is said to be a derangement if it has no fixed points on Ω. From the orbit counting lemma, it follows that G ...contains a derangement, and in fact G contains a derangement of prime power order by a theorem of Fein, Kantor and Schacher. However, there are groups with no derangements of prime order; these are the so-called elusive groups and they have been widely studied in recent years. Extending this notion, we say that G is almost elusive if it contains a unique conjugacy class of derangements of prime order. In this paper we first prove that every quasiprimitive almost elusive group is either almost simple or 2-transitive of affine type. We then classify all the almost elusive groups that are almost simple and primitive with socle an alternating group, a sporadic group, or a rank one group of Lie type.
We determine the 2-fusion systems of J-component type in which the centralizer of some fully centralized involution has a maximal J-component that is the 2-fusion system of U3(3).
Let Γ be a countable group and (X,Γ) a compact topological dynamical system. We study the question of the existence of an intermediate C⁎-subalgebra ACr⁎(Γ)<A<C(X)⋊rΓ, which is not of the form ...A=C(Y)⋊rΓ, corresponding to a factor map (X,Γ)→(Y,Γ). Here Cr⁎(Γ) is the reduced C⁎-algebra of Γ and C(X)⋊rΓ is the reduced C⁎-crossed-product of (X,Γ). Our main results are: (1) For Γ which is not C⁎-simple, when (X,Γ) admits a Γ-invariant probability measure, then such a sub-algebra always exists. (2) For Γ=Z and (X,Γ) an irrational rotation of the circle X=R/Z, we give a full description of all these non-crossed-product subalgebras.
The spread of a finite group Burness, Timothy C.; Guralnick, Robert M.; Harper, Scott
Annals of mathematics,
03/2021, Letnik:
193, Številka:
2
Journal Article
Odprti dostop
A group G is said to be 3/2-generated if every nontrivial
element belongs to a generating pair. It is easy to see that if
G has this property, then every proper quotient of
G is cyclic. In this paper ...we prove that the converse is
true for finite groups, which settles a conjecture of Breuer, Guralnick and
Kantor from 2008. In fact, we prove a much stronger result, which solves a
problem posed by Brenner and Wiegold in 1975. Namely, if G is a
finite group and every proper quotient of G is cyclic, then for
any pair of nontrivial elements x
1,
x
2 ϵ G, there exists
y ϵ G such that G =
⟨x
1, y⟩ =
⟨x
2, y⟩. In other words,
s(G) ⩾ 2, where
s(G) is the spread of G.
Moreover, if u(G) denotes the more restrictive
uniform spread of G, then we can completely characterise the
finite groups G with u(G) = 0
and u(G) = 1. To prove these results, we first
establish a reduction to almost simple groups. For simple groups, the result was
proved by Guralnick and Kantor in 2000 using probabilistic methods, and since
then the almost simple groups have been the subject of several papers. By
combining our reduction theorem and this earlier work, it remains to handle the
groups with socle an exceptional group of Lie type, and this is the case we
treat in this paper.
We study C⁎-irreducibility of inclusions of reduced twisted group C⁎-algebras and of reduced group C⁎-algebras. We characterize C⁎-irreducibility in the case of an inclusion arising from a normal ...subgroup, and exhibit many new examples of C⁎-irreducible inclusions.
We prove that if L a set of generators of L grows, i.e., \vert A^3\vert > \vert A\vert^{1+\varepsilon } \varepsilon , or A^3=L A generalization of our proof yields the following. Let A ...SL(n,\mathbb{F}) \mathbb{F} \big \vert A^3\big \vert\le \mathcal {K}\vert A\vert can be covered by \mathcal {K}^m SL(n,\mathbb{F}), where m.
McKay graphs for alternating and classical groups Liebeck, Martin W.; Shalev, Aner; Tiep, Pham Huu
Transactions of the American Mathematical Society,
08/2021, Letnik:
374, Številka:
8
Journal Article
Recenzirano
Odprti dostop
Let G be a finite group, and \alpha a nontrivial character of G. The McKay graph \mathcal {M}(G,\alpha ) has the irreducible characters of G as vertices, with an edge from \chi _1 to \chi _2 if \chi ..._2 is a constituent of \alpha \chi _1. We study the diameters of McKay graphs for finite simple groups G. For alternating groups G = \mathsf {A}_n, we prove a conjecture made in another work by the authors: there is an absolute constant C such that \mathrm {diam} {\mathcal M}(G,\alpha ) \le C\frac {\log |G|}{\log \alpha (1)} for all nontrivial irreducible characters \alpha of G. Also for classical groups of symplectic or orthogonal type of rank r, we establish a linear upper bound Cr on the diameters of all nontrivial McKay graphs. Finally, we provide some sufficient conditions for a product \chi _1\chi _2\cdots \chi _l of irreducible characters of some finite simple groups G to contain all irreducible characters of G as constituents.
We prove that if G is a sufficiently large finite almost simple group of Lie type, then given a fixed nontrivial element x∈G and a coset of G modulo its socle, the probability that x and a random ...element of the coset generate a subgroup containing the socle is uniformly bounded away from 0 (and goes to 1 if the field size goes to ∞). This is new even if G is simple. Together with results of Lucchini and Burness–Guralnick–Harper, this proves a conjecture of Lucchini and has an application to profinite groups. A key step in the proof is the determination of the limits for the proportion of elements in a classical group which fix no subspace of any bounded dimension.
PDF
