Volume 23, Issue 3

G. Navarro raised the question of when two vertices of two indecomposable modules over a finite group algebra generate a Sylow p -subgroup. The present note provides a sufficient criterion for this to happen. This generalises a result by Navarro for simple modules over finite p -solvable groups, which is the main motivation for this note.

Let G be a finite transitive permutation group of degree n , with point stabilizer H≠1{H\neq 1} and permutation character π. For every positive integer t , we consider the generalized character ψt=ρG-t⁢(π-1G){\psi_{t}=\rho_{G}-t(\pi-1_{G})}, where ρG{\rho_{G}} is the regular character of G and 1G{1_{G}} the 1-character. We give necessary and sufficient conditions on t (and G ) which guarantee that ψt{\psi_{t}} is a character of G . A necessary condition is that t≤min⁡{n-1,|H|}{t\leq\min\{n-1,\lvert H\rvert\}}, and it turns out that ψt{\psi_{t}} is a character of G for t=n-1{t=n-1} resp. t=|H|{t=\lvert H\rvert} precisely when G is 2-transitive resp. a Frobenius group.

The Jordan–Hölder theorem is a general term given to a collection of theorems about maximal chains in suitably nice lattices. For example, the well-known Jordan–Hölder type theorem for chief series of finite groups has been rather useful in studying the structure of finite groups. In this paper, we present a Jordan–Hölder type theorem for supercharacter theories of finite groups, which generalizes the one for chief series of finite groups.

We prove that each exotic fusion system ℱ{\mathcal{F}} on a Sylow p -subgroup of G2⁢(p){G_{2}(p)} for an odd prime p with 𝒪p⁢(ℱ)=1{\mathcal{O}_{p}(\mathcal{F})=1} is block-exotic. This gives evidence for the conjecture that each exotic fusion system is block-exotic. We prove two reduction theorems for block-realisable fusion systems.

Let R be a finite unital commutative ring. We introduce a new class of finite groups, which we call hereditary groups over R . Our main result states that if G is a hereditary group over R , then a unital algebra isomorphism between group algebras R⁢G≅R⁢H{RG\cong RH} implies a group isomorphism G≅H{G\cong H} for every finite group H . As application, we study the modular isomorphism problem, which is the isomorphism problem for finite p -groups over R=𝔽p{R=\mathbb{F}_{p}}, where 𝔽p{\mathbb{F}_{p}} is the field of p elements. We prove that a finite p -group G is a hereditary group over 𝔽p{\mathbb{F}_{p}} provided G is abelian, G is of class two and exponent p , or G is of class two and exponent four. These yield new proofs for the theorems by Deskins and Passi–Sehgal.

We refer to the set of the orders of elements of a finite group as its spectrum and say that groups are isospectral if their spectra coincide. We prove that, except for one specific case, the solvable radical of a nonsolvable finite group isospectral to a finite simple group is nilpotent.

Héthelyi and Külshammer showed that the number of conjugacy classes k⁢(G){k(G)} of any solvable finite group G whose order is divisible by the square of a prime p is at least (49⁢p+1)/60{(49p+1)/60}. Here an asymptotic generalization of this result is established. It is proved that there exists a constant c>0{c>0} such that, for any finite group G whose order is divisible by the square of a prime p , we have k⁢(G)≥c⁢p{k(G)\geq cp}.

Let κ be a characteristic p finite field of q elements and 𝔑κ{\mathfrak{N}_{\kappa}} the Nottingham group over κ. Lubin associated to every conjugacy class of torsion element of 𝔑κ{\mathfrak{N}_{\kappa}} a type. We establish an upper bound B⁢(q;l,m){B(q;l,m)} on the number of conjugacy classes of order p2{p^{2}} torsion elements u of 𝔑κ{\mathfrak{N}_{\kappa}} of type 〈l,m〉{\langle l,m\rangle}. In the case where l<p{l<p}, the bound B⁢(q;l,m){B(q;l,m)} is the exact number of conjugacy classes. Moreover, we give a criterion on when u and un{u^{n}} are conjugate.

Let n be a positive integer. We say that a group G is an (n+12){(n+\frac{1}{2})}-Engel group if it satisfies the law [x,yn,x]=1{[x,{}_{n}y,x]=1}. The variety of (n+12){(n+\frac{1}{2})}-Engel groups lies between the varieties of n -Engel groups and (n+1){(n+1)}-Engel groups. In this paper, we study these groups, and in particular, we prove that all (4+12){(4+\frac{1}{2})}-Engel {2,3}{\{2,3\}}-groups are locally nilpotent. We also show that if G is a (4+12){(4+\frac{1}{2})}-Engel p -group, where p≥5{p\geq 5} is a prime, then Gp{G^{p}} is locally nilpotent.

If 𝔛{\mathfrak{X}} is a class of groups, define a sequence of group classes 𝔛1,𝔛2,…,𝔛k,…{\mathfrak{X}_{1},\mathfrak{X}_{2},\ldots,\mathfrak{X}_{k},\ldots} by putting 𝔛1=𝔛{\mathfrak{X}_{1}=\mathfrak{X}} and choosing 𝔛k+1{\mathfrak{X}_{k+1}} as the class of all groups whose non-permutable subgroups belong to 𝔛k{\mathfrak{X}_{k}}. In particular, if 𝔄{\mathfrak{A}} is the class of abelian groups, 𝔄2{\mathfrak{A}_{2}} is the class of quasimetahamiltonian groups, i.e. groups whose non-permutable subgroups are abelian. The aim of this paper is to study the structure of 𝔛k{\mathfrak{X}_{k}}-groups, with special emphasis on the case 𝔛=𝔄{\mathfrak{X}=\mathfrak{A}}. Among other results, it will also be proved that a group has a finite normal subgroup with quasihamiltonian quotient if and only if it is locally graded and belongs to 𝔄k{\mathfrak{A}_{k}} for some positive integer k .

For certain uncountable cardinals κ, we produce a group of cardinality κ which is freely indecomposable, strongly κ-free, and whose abelianization is free abelian of rank κ. The construction takes place in Gödel’s constructible universe L . This strengthens an earlier result of Eklof and Mekler.

A group G has the R∞{R_{\infty}}-property if the number R⁢(φ){R(\varphi)} of twisted conjugacy classes is infinite for any automorphism φ of G . For such a group G , the R∞{R_{\infty}}-nilpotency degree is the least integer c such that G/γc+1⁢(G){G/\gamma_{c+1}(G)} still has the R∞{R_{\infty}}-property. In this paper, we determine the R∞{R_{\infty}}-nilpotency degree of all Baumslag–Solitar groups.