Volume 22, Issue 5

Let F be Thompson’s group. We prove that F satisfies laws with constants but does not satisfy any semigroup laws with constants. We also show that there exist no transferable laws with constants in F .

A machinery developed by the second author produces a rich family of unitary representations of the Thompson groups F , T and V . We use it to give direct proofs of two previously known results. First, we exhibit a unitary representation of V that has an almost invariant vector but no nonzero [F,F]{[F,F]}-invariant vectors reproving and extending Reznikoff’s result that any intermediate subgroup between the commutator subgroup of F and V does not have Kazhdan’s property (T) (though Reznikoff proved it for subgroups of T ). Second, we construct a one parameter family interpolating between the trivial and the left regular representations of V . We exhibit a net of coefficients for those representations which vanish at infinity on T and converge to 1 thus reproving that T has the Haagerup property after Farley who further proved that V has this property.

A comprehensive account is given of the theory of metanilpotent groups with the minimal condition on normal subgroups. After reviewing classical material, many new results are established relating to the Fitting subgroup, the Hirsch–Plotkin radical, the Frattini subgroup, splitting and conjugacy, the Schur multiplier, Sylow structure and the maximal subgroups. Module theoretic and homological methods are used throughout.

We establish conditions under which the fundamental group of a graph of finite p -groups is necessarily residually p -finite. The technique of proof is independent of previously established results of this type, and the result is also valid for infinite graphs of groups.

We use character theory and character estimates to show that the number of T2{T_{2}}-systems of the alternating group An{A_{n}} is at least 18⁢n⁢3⁢exp⁡(2⁢π6⁢n1/2)⁢(1+o⁢(1)).\frac{1}{8n\sqrt{3}}\exp\biggl{(}\frac{2\pi}{\sqrt{6}}n^{1/2}\biggr{)}(1+o(1)). Applying this result, we obtain a lower bound for the number of connected components of the product replacement graph Γ2⁢(An){\Gamma_{2}(A_{n})}.

Let Irr⁡(G){\operatorname{Irr}(G)} denote the set of complex irreducible characters of a finite group G , and let cd⁡(G){\operatorname{cd}(G)} be the set of degrees of the members of Irr⁡(G){\operatorname{Irr}(G)}. For positive integers k and l , we say that the finite group G has the property 𝒫kl{\mathcal{P}^{l}_{k}} if, for any distinct degrees a1,a2,…,ak∈cd⁡(G){a_{1},a_{2},\dots,a_{k}\in\operatorname{cd}(G)}, the total number of (not necessarily different) prime divisors of the greatest common divisor gcd⁡(a1,a2,…,ak){\gcd(a_{1},a_{2},\dots,a_{k})} is at most l-1{l-1}. In this paper, we classify all finite almost simple groups satisfying the property 𝒫32{\mathcal{P}_{3}^{2}}. As a consequence of our classification, we show that if G is an almost simple group satisfying 𝒫32{\mathcal{P}_{3}^{2}}, then |cd⁡(G)|⩽8{\lvert\operatorname{cd}(G)\rvert\leqslant 8}.

Let V be a vector space over a field k . We call a subgroup G⊂GL⁢(V){G\subset\mathrm{GL}(V)} a fixed-point subgroup if det⁡(1-g)=0{\det(1-g)=0} for all g∈G{g\in G}. Let q be a power of a prime. In this paper, we classify the fixed-point subgroups of GL3⁢(q){\mathrm{GL}_{3}(q)}.

Let G be a finite group, and let 𝔉{\mathfrak{F}} be a class of groups. A chief factor H/K{H/K} of G is said to be 𝔉{\mathfrak{F}}-central (in G ) if the semidirect product (H/K)⋊(G/CG⁢(H/K))∈𝔉{(H/K)\rtimes(G/C_{G}(H/K))\in\mathfrak{F}}. We say that a subgroup A of G is 𝔉{\mathfrak{F}}-normal in G if every chief factor H/K{H/K} of G between AG{A_{G}} and AG{A^{G}} is 𝔉{\mathfrak{F}}-central in G and 𝔉{\mathfrak{F}}-abnormal in G if V is not 𝔉{\mathfrak{F}}-normal in W for every two subgroups V<W{V<W} of G such that A≤V{A\leq V}. We give a description of finite groups in which every subgroup is either 𝔉{\mathfrak{F}}-normal or 𝔉{\mathfrak{F}}-abnormal.

Let acd⁡(G){\operatorname{acd}(G)} and acs⁡(G){\operatorname{acs}(G)} denote the average irreducible character degree and the average conjugacy class size, respectively, of a finite group G . The object of this paper is to prove that if acd⁡(G)<2⁢(p+1)/(p+3)\operatorname{acd}(G)<2(p+1)/(p+3), then G=Op⁢(G)×Op′⁢(G)G=O_{p}(G)\times O_{{p^{\prime}}}(G), and that if acs⁡(G)<4⁢p/(p+3)\operatorname{acs}(G)<4p/(p\kern-1.0pt+\kern-1.0pt3), then G=Op⁢(G)×Op′⁢(G)G=O_{p}(G)\kern-1.0pt\times\kern-1.0ptO_{{p^{\prime}}}(G) with Op⁢(G){O_{p}(G)} abelian, where p is a prime.

Let G be a finite group and p a prime. Let cl⁡(G){\operatorname{cl}(G)} be the set of conjugacy classes of G , and let eclp⁡(G){\operatorname{ecl}_{p}(G)} be the largest integer such that peclp⁡(G){p^{\operatorname{ecl}_{p}(G)}} divides |C|{|C|} for some C∈cl⁡(G){C\in\operatorname{cl}(G)}. We show that if p≥3{p\geq 3} and eclp⁡(G)=1{\operatorname{ecl}_{p}(G)=1}, then |G⁢:⁢Op⁢(G)|p≤p3{\lvert G\mskip 1.0mu \mathord{:}\mskip 1.0mu O_{p}(G)\rvert_{p}\leq p^{3}}. This improves the main result of Y. Yang and G. Qian, On p -parts of conjugacy class sizes of finite groups, Bull. Aust. Math. Soc. 97 2018, 3, 406–411.

For a given prime p , a finite group G is said to be a 𝒞~p{\widetilde{\mathcal{C}}_{p}}-group if every cyclic p -subgroup of G is self-normalizing in its subnormal closure. In this paper, we get some descriptions of 𝒞~p{\widetilde{\mathcal{C}}_{p}}-groups, show that the class of 𝒞~p{\widetilde{\mathcal{C}}_{p}}-groups is a subgroup-closed formation and that Op′⁢(G){O^{p^{\prime}}(G)} is a solvable p -nilpotent group for every 𝒞~p{\widetilde{\mathcal{C}}_{p}}-group G . We also prove that if a finite group G is a 𝒞~p{\widetilde{\mathcal{C}}_{p}}-group for all primes p , then every subgroup of G is self-normalizing in its subnormal closure.

H. J. Zassenhaus conjectured that any unit of finite order and augmentation 1 in the integral group ring ℤ⁢G{\mathbb{Z}G} of a finite group G is conjugate in the rational group algebra ℚ⁢G{\mathbb{Q}G} to an element of G . We prove the Zassenhaus conjecture for the groups SL⁢(2,p){\mathrm{SL}(2,p)} and SL⁢(2,p2){\mathrm{SL}(2,p^{2})} with p a prime number. This is the first infinite family of non-solvable groups for which the Zassenhaus conjecture has been proved. We also prove that if G=SL⁢(2,pf){G=\mathrm{SL}(2,p^{f})}, with f arbitrary and u is a torsion unit of ℤ⁢G{\mathbb{Z}G} with augmentation 1 and order coprime with p , then u is conjugate in ℚ⁢G{\mathbb{Q}G} to an element of G . By known results, this reduces the proof of the Zassenhaus conjecture for these groups to proving that every unit of ℤ⁢G{\mathbb{Z}G} of order a multiple of p and augmentation 1 has order actually equal to p .