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We show that the only finite quasi-simple non-abelian groups that can faithfully act on rationally connected threefolds are the following groups:
${\mathfrak{A}}_5$
,
${\text{PSL}}_2(\textbf{F}_7)$
,
${\mathfrak{A}}_6$
,
${\text{SL}}_2(\textbf{F}_8)$
,
${\mathfrak{A}}_7$
,
${\text{PSp}}_4(\textbf{F}_3)$
,
${\text{SL}}_2(\textbf{F}_{7})$
,
$2.{\mathfrak{A}}_5$
,
$2.{\mathfrak{A}}_6$
,
$3.{\mathfrak{A}}_6$
or
$6.{\mathfrak{A}}_6$
. All of these groups with a possible exception of
$2.{\mathfrak{A}}_6$
and
$6.{\mathfrak{A}}_6$
indeed act on some rationally connected threefolds.
In their renowned paper (2011, Inventiones Mathematicae 184, 591–627), I. Vollaard and T. Wedhorn defined a stratification on the special fiber of the unitary unramified PEL Rapoport–Zink space with signature $(1,n-1)$. They constructed an isomorphism between the closure of a stratum, called a closed Bruhat–Tits stratum, and a Deligne–Lusztig variety which is not of classical type. In this paper, we describe the $\ell $-adic cohomology groups over $\overline {{\mathbb Q}_{\ell }}$ of these Deligne–Lusztig varieties, where $\ell \not = p$. The computations involve the spectral sequence associated with the Ekedahl–Oort stratification of a closed Bruhat–Tits stratum, which translates into a stratification by Coxeter varieties whose cohomology is known. Eventually, we find out that the irreducible representations of the finite unitary group which appear inside the cohomology contribute to only two different unipotent Harish-Chandra series, one of them belonging to the principal series.
In this paper, we study the supercharacter theories of elementary abelian $p$-groups of order $p^{2}$. We show that the supercharacter theories that arise from the direct product construction and the $\ast$-product construction can be obtained from automorphisms. We also prove that any supercharacter theory of an elementary abelian $p$-group of order $p^{2}$ that has a non-identity superclass of size $1$ or a non-principal linear supercharacter must come from either a $\ast$-product or a direct product. Although we are unable to prove results for general primes, we do compute all of the supercharacter theories when $p = 2,\, 3,\, 5$, and based on these computations along with particular computations for larger primes, we make several conjectures for a general prime $p$.
The codegree of an irreducible character
$\chi $
of a finite group G is
$|G : \ker \chi |/\chi (1)$
. We show that the Ree group
${}^2G_2(q)$
, where
$q = 3^{2f+1}$
, is determined up to isomorphism by its set of codegrees.
We study the compatibility of the action of the DAHA of type GL with two inverse systems of polynomial rings obtained from the standard Laurent polynomial representations. In both cases, the crucial analysis is that of the compatibility of the action of the Cherednik operators. Each case leads to a representation of a limit structure (the +/– stable limit DAHA) on a space of almost symmetric polynomials in infinitely many variables (the standard representation). As an application, we show that the defining representation of the double Dyck path algebra arises from the standard representation of the +stable limit DAHA.
Gagola and Lewis [‘A character theoretic condition characterizing nilpotent groups’, Comm. Algebra27 (1999), 1053–1056] proved that a finite group G is nilpotent if and only if
$\chi (1)^{2}$
divides
$\lvert G:\textrm {ker}\,\chi \rvert $
for every irreducible character
$\chi $
of G. The theorem was later generalised by using monolithic characters. We generalise the theorem further considering only strongly monolithic characters. We also give some criteria for solvability and nilpotency of finite groups by their strongly monolithic characters.
The Hanna Neumann conjecture is a statement about the rank of the intersection of two finitely generated subgroups of a free group. The conjecture was posed by Hanna Neumann in 1957. In 2011, a strengthened version of the conjecture was proved independently by Joel Friedman and by Igor Mineyev. In this paper we show that the strengthened Hanna Neumann conjecture holds not only in free groups but also in non-solvable surface groups. In addition, we show that a retract in a free group and in a surface group is inert. This implies the Dicks–Ventura inertia conjecture for free and surface groups.
We describe finite soluble nonnilpotent groups in which every minimal nonnilpotent subgroup is abnormal. We also show that if G is a nonsoluble finite group in which every minimal nonnilpotent subgroup is abnormal, then G is quasisimple and
$Z(G)$
is cyclic of order
$|Z(G)|\in \{1, 2, 3, 4\}$
.
Given a finite group G, we denote by
$L(G)$
the subgroup lattice of G and by
${\cal CD}(G)$
the Chermak–Delgado lattice of G. In this note, we determine the finite groups G such that
$|{\cal CD}(G)|=|L(G)|-k$
, for
$k=1,2$
.
In this note, we investigate some products of subgroups and vanishing conjugacy class sizes of finite groups. We prove some supersolubility criteria for groups with restrictions on the vanishing conjugacy class sizes of their subgroups.
Let K be an infinite field of characteristic
$p>0$
and let
$\lambda, \mu$
be partitions, where
$\mu$
has two parts. We find sufficient arithmetic conditions on
$p, \lambda, \mu$
for the existence of a nonzero homomorphism
$\Delta(\lambda) \to \Delta (\mu)$
of Weyl modules for the general linear group
$GL_n(K)$
. Also, for each p we find sufficient conditions so that the corresponding homomorphism spaces have dimension at least 2.
Let
$\mathcal {C}_n =\left [\chi _{\lambda }(\mu )\right ]_{\lambda , \mu }$
be the character table for
$S_n,$
where the indices
$\lambda $
and
$\mu $
run over the
$p(n)$
many integer partitions of
$n.$
In this note, we study
$Z_{\ell }(n),$
the number of zero entries
$\chi _{\lambda }(\mu )$
in
$\mathcal {C}_n,$
where
$\lambda $
is an
$\ell $
-core partition of
$n.$
For every prime
$\ell \geq 5,$
we prove an asymptotic formula of the form
where
$\sigma _{\ell }(n)$
is a twisted Legendre symbol divisor function,
$\delta _{\ell }:=(\ell ^2-1)/24,$
and
$1/\alpha _{\ell }>0$
is a normalization of the Dirichlet L-value
$L\left (\left ( \frac {\cdot }{\ell } \right ),\frac {\ell -1}{2}\right ).$
For primes
$\ell $
and
$n>\ell ^6/24,$
we show that
$\chi _{\lambda }(\mu )=0$
whenever
$\lambda $
and
$\mu $
are both
$\ell $
-cores. Furthermore, if
$Z^*_{\ell }(n)$
is the number of zero entries indexed by two
$\ell $
-cores, then, for
$\ell \geq 5$
, we obtain the asymptotic
Let G be a p-adic classical group. The representations in a given Bernstein component can be viewed as modules for the corresponding Hecke algebra—the endomorphism algebra of a pro-generator of the given component. Using Heiermann’s construction of these algebras, we describe the Bernstein components of the Gelfand–Graev representation for $G=\mathrm {SO}(2n+1)$, $\mathrm {Sp}(2n)$, and $\mathrm {O}(2n)$.
The Chermak–Delgado lattice of a finite group G is a self-dual sublattice of the subgroup lattice of G. In this paper, we prove that, for any finite abelian group A, there exists a finite group G such that the Chermak–Delgado lattice of G is a subgroup lattice of A.
We present a family of counterexamples to a question proposed recently by Moretó concerning the character codegrees and the element orders of a finite solvable group.
Let
$\eta (G)$
be the number of conjugacy classes of maximal cyclic subgroups of G. We prove that if G is a p-group of order
$p^n$
and nilpotence class l, then
$\eta (G)$
is bounded below by a linear function in
$n/l$
.