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We show that if w is a multilinear commutator word and G a finite group in which every metanilpotent subgroup generated by w-values is of rank at most r, then the rank of the verbal subgroup $w(G)$ is bounded in terms of r and w only. In the case where G is soluble, we obtain a better result: if G is a finite soluble group in which every nilpotent subgroup generated by w-values is of rank at most r, then the rank of $w(G)$ is at most $r+1$.
We present a sufficient condition for the $kG$-Scott module with vertex $P$ to remain indecomposable under the Brauer construction for any subgroup $Q$ of $P$ as $k[Q\,C_G(Q)]$-module, where $k$ is a field of characteristic $2$, and $P$ is a semidihedral $2$-subgroup of a finite group $G$. This generalizes results for the cases where $P$ is abelian or dihedral. The Brauer indecomposability is defined by R. Kessar, N. Kunugi and N. Mitsuhashi. The motivation of this paper is the fact that the Brauer indecomposability of a $p$-permutation bimodule (where $p$ is a prime) is one of the key steps in order to obtain a splendid stable equivalence of Morita type by making use of the gluing method due to Broué, Rickard, Linckelmann and Rouquier, that then can possibly be lifted to a splendid derived (splendid Morita) equivalence.
We prove that the invariably generating graph of a finite group can have an arbitrarily large number of connected components with at least two vertices.
In this note, we compute the centers of the categories of tilting modules for G = SL2 in prime characteristic, of tilting modules for the corresponding quantum group at a complex root of unity, and of projective GgT-modules when g = 1, 2.
Let $F$ be a $p$-adic field and choose $k$ an algebraic closure of $\mathbb{F}_{\ell }$, with $\ell$ different from $p$. We define “nilpotent lifts” of irreducible generic $k$-representations of $GL_{n}(F)$, which take coefficients in Artin local $k$-algebras. We show that an irreducible generic $\ell$-modular representation $\unicode[STIX]{x1D70B}$ of $GL_{n}(F)$ is uniquely determined by its collection of Rankin–Selberg gamma factors $\unicode[STIX]{x1D6FE}(\unicode[STIX]{x1D70B}\times \widetilde{\unicode[STIX]{x1D70F}},X,\unicode[STIX]{x1D713})$ as $\widetilde{\unicode[STIX]{x1D70F}}$ varies over nilpotent lifts of irreducible generic $k$-representations $\unicode[STIX]{x1D70F}$ of $GL_{t}(F)$ for $t=1,\ldots ,\lfloor \frac{n}{2}\rfloor$. This gives a characterization of the mod-$\ell$ local Langlands correspondence in terms of gamma factors, assuming it can be extended to a surjective local Langlands correspondence on nilpotent lifts.
Let G be a finite group, let
${\text{Irr}}(G)$ be the set of all irreducible complex characters of G and let
$\chi \in {\text{Irr}}(G)$. Define the codegrees,
${\text{cod}}(\chi ) = |G: {\text{ker}}\chi |/\chi (1)$ and
${\text{cod}}(G) = \{{\text{cod}}(\chi ) \mid \chi \in {\text{Irr}}(G)\} $. We show that the simple group
${\text{PSL}}(2,q)$, for a prime power
$q>3$, is uniquely determined by the set of its codegrees.
Let G be a finite group and $\psi (G) = \sum _{g \in G} o(g)$, where $o(g)$ denotes the order of $g \in G$. There are many results on the influence of this function on the structure of a finite group G.
In this paper, as the main result, we answer a conjecture of Tărnăuceanu. In fact, we prove that if G is a group of order n and $\psi (G)>31\psi (C_n)/77$, where $C_n$ is the cyclic group of order n, then G is supersolvable. Also, we prove that if G is not a supersolvable group of order n and $\psi (G) = 31\psi (C_n)/77$, then $G\cong A_4 \times C_m$, where $(m, 6)=1$.
Finally, Herzog et al. in (2018, J. Algebra, 511, 215–226) posed the following conjecture: If $H\leq G$, then $\psi (G) \unicode[stix]{x02A7D} \psi (H) |G:H|^2$. By an example, we show that this conjecture is not satisfied in general.
A key ingredient in the Taylor–Wiles proof of Fermat’s last theorem is the classical Ihara lemma, which is used to raise the modularity property between some congruent Galois representations. In their work on Sato and Tate, Clozel, Harris and Taylor proposed a generalisation of the Ihara lemma in higher dimension for some similitude groups. The main aim of this paper is to prove some new instances of this generalised Ihara lemma by considering some particular non-pseudo-Eisenstein maximal ideals of unramified Hecke algebras. As a consequence, we prove a level-raising statement.
Let G be a finite group and
$\chi $ be a character of G. The codegree of
$\chi $ is
$\text{codeg} (\chi ) ={|G: \ker \chi |}/{\chi (1)}$. We write
$\pi (G)$ for the set of prime divisors of
$|G|$,
$\pi (\text{codeg} (\chi ))$ for the set of prime divisors of
$\text{codeg} (\chi )$ and
$\sigma (\text{codeg} (G))= \max \{|\pi (\text{codeg} (\chi ))| : \chi \in {\textrm {Irr}}(G)\}$. We show that
$|\pi (G)| \leq ({23}/{3}) \sigma (\text{codeg} (G))$. This improves the main result of Yang and Qian [‘The analog of Huppert's conjecture on character codegrees’, J. Algebra478 (2017), 215–219].
A rigid automorphism of a linking system is an automorphism that restricts to the identity on the Sylow subgroup. A rigid inner automorphism is conjugation by an element in the center of the Sylow subgroup. At odd primes, it is known that each rigid automorphism of a centric linking system is inner. We prove that the group of rigid outer automorphisms of a linking system at the prime $2$ is elementary abelian and that it splits over the subgroup of rigid inner automorphisms. In a second result, we show that if an automorphism of a finite group G restricts to the identity on the centric linking system for G, then it is of
$p'$-order modulo the group of inner automorphisms, provided G has no nontrivial normal
$p'$-subgroups. We present two applications of this last result, one to tame fusion systems.
We study the fields of values of the irreducible characters of a finite group of degree not divisible by a prime p. In the case where $p=2$, we fully characterise these fields. In order to accomplish this, we generalise the main result of [ILNT] to higher irrationalities. We do the same for odd primes, except that in this case the analogous results hold modulo a simple-to-state conjecture on the character values of quasi-simple groups.
A long standing problem, which has its roots in low-dimensional homotopy theory, is to classify all finite groups G for which the integral group ring ℤG has stably free cancellation (SFC). We extend results of R. G. Swan by giving a condition for SFC and use this to show that ℤG has SFC provided at most one copy of the quaternions ℍ occurs in the Wedderburn decomposition of the real group ring ℝG. This generalises the Eichler condition in the case of integral group rings.
For a finite group G, let
$\Delta (G)$
denote the character graph built on the set of degrees of the irreducible complex characters of G. A perfect graph is a graph
$\Gamma $
in which the chromatic number of every induced subgraph
$\Delta $
of
$\Gamma $
equals the clique number of
$\Delta $
. We show that the character graph
$\Delta (G)$
of a finite group G is always a perfect graph. We also prove that the chromatic number of the complement of
$\Delta (G)$
is at most three.
We establish the inductive blockwise Alperin weight condition for simple groups of Lie type
$\mathsf C$
and the bad prime
$2$
. As a main step, we derive a labelling set for the irreducible
$2$
-Brauer characters of the finite symplectic groups
$\operatorname {Sp}_{2n}(q)$
(with odd q), together with the action of automorphisms. As a further important ingredient, we prove a Jordan decomposition for weights.
We consider the graph
$\Gamma _{\text {virt}}(G)$
whose vertices are the elements of a finitely generated profinite group G and where two vertices x and y are adjacent if and only if they topologically generate an open subgroup of G. We investigate the connectivity of the graph
$\Delta _{\text {virt}}(G)$
obtained from
$\Gamma _{\text {virt}}(G)$
by removing its isolated vertices. In particular, we prove that for every positive integer t, there exists a finitely generated prosoluble group G with the property that
$\Delta _{\operatorname {\mathrm {virt}}}(G)$
has precisely t connected components. Moreover, we study the graph
$\widetilde \Gamma _{\operatorname {\mathrm {virt}}}(G)$
, whose vertices are again the elements of G and where two vertices are adjacent if and only if there exists a minimal generating set of G containing them. In this case, we prove that the subgraph
$\widetilde \Delta _{\operatorname {\mathrm {virt}}}(G)$
obtained removing the isolated vertices is connected and has diameter at most 3.
In 2014, Baumslag and Wiegold proved that a finite group G is nilpotent if and only if o(xy) = o(x)o(y) for every x, y ∈ G with (o(x), o(y)) = 1. This has led to a number of results that characterize the nilpotence of a group (or the existence of nilpotent Hall subgroups, or the existence of normal Hall subgroups) in terms of prime divisors of element orders. Here, we look at these results with a new twist. The first of our main results asserts that G is nilpotent if and only if o(xy) ⩽ o(x)o(y) for every x, y ∈ G of prime power order with (o(x), o(y)) = 1. As an immediate consequence, we recover the Baumslag–Wiegold theorem. The proof of this result is elementary. We prove some variations of this result that depend on the classification of finite simple groups.
We present several results on the connectivity of McKay quivers of finite-dimensional complex representations of finite groups, with no restriction on the faithfulness or self-duality of the representations. We give examples of McKay quivers, as well as quivers that cannot arise as McKay quivers, and discuss a necessary and sufficient condition for two finite groups to share a connected McKay quiver.
Linckelmann and Murphy have classified the Morita equivalence classes of p-blocks of finite groups whose basic algebra has dimension at most
$12$
. We extend their classification to dimension
$13$
and
$14$
. As predicted by Donovan’s conjecture, we obtain only finitely many such Morita equivalence classes.