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Isaacs and Seitz conjectured that the derived length of a finite solvable group
$G$
is bounded by the cardinality of the set of all irreducible character degrees of
$G$
. We prove that the conjecture holds for
$G$
if the degrees of nonlinear monolithic characters of
$G$
having the same kernels are distinct. Also, we show that the conjecture is true when
$G$
has at most three nonlinear monolithic characters. We give some sufficient conditions for the inequality related to monolithic characters or real-valued irreducible characters of
$G$
when the commutator subgroup of
$G$
is supersolvable.
We prove Bogolyubov–Ruzsa-type results for finite subsets of groups with small tripling, |A3| ≤ O(|A|), or small alternation, |AA−1A| ≤ O(|A|). As applications, we obtain a qualitative analogue of Bogolyubov’s lemma for dense sets in arbitrary finite groups, as well as a quantitative arithmetic regularity lemma for sets of bounded VC-dimension in finite groups of bounded exponent. The latter result generalizes the abelian case, due to Alon, Fox and Zhao, and gives a quantitative version of previous work of the author, Pillay and Terry.
For a finite group
$G$
, define
$l(G)=(\prod _{g\in G}o(g))^{1/|G|}/|G|$
, where
$o(g)$
denotes the order of
$g\in G$
. We prove that if
$l(G)>l(A_{5}),l(G)>l(A_{4}),l(G)>l(S_{3}),l(G)>l(Q_{8})$
or
$l(G)>l(C_{2}\times C_{2})$
, then
$G$
is solvable, supersolvable, nilpotent, abelian or cyclic, respectively.
Let
$F$
be a totally real field in which
$p$
is unramified. Let
$\overline{r}:G_{F}\rightarrow \text{GL}_{2}(\overline{\mathbf{F}}_{p})$
be a modular Galois representation that satisfies the Taylor–Wiles hypotheses and is tamely ramified and generic at a place
$v$
above
$p$
. Let
$\mathfrak{m}$
be the corresponding Hecke eigensystem. We describe the
$\mathfrak{m}$
-torsion in the
$\text{mod}\,p$
cohomology of Shimura curves with full congruence level at
$v$
as a
$\text{GL}_{2}(k_{v})$
-representation. In particular, it only depends on
$\overline{r}|_{I_{F_{v}}}$
and its Jordan–Hölder factors appear with multiplicity one. The main ingredients are a description of the submodule structure for generic
$\text{GL}_{2}(\mathbf{F}_{q})$
-projective envelopes and the multiplicity one results of Emerton, Gee and Savitt [Lattices in the cohomology of Shimura curves, Invent. Math.200(1) (2015), 1–96].
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.
We present a complete list of groups
$G$
and fields
$F$
for which: (i) the group of normalized units
$V(FG)$
of the group algebra
$FG$
is locally nilpotent; (ii) the set of nontrivial nilpotent elements of
$FG$
is finite and nonempty, and
$V(FG)$
is an Engel group.
Let
$G$
be a finite group and
$p$
be an odd prime. We show that if
$\mathbf{O}_{p}(G)=1$
and
$p^{2}$
does not divide every irreducible
$p$
-Brauer character degree of
$G$
, then
$|G|_{p}$
is bounded by
$p^{3}$
when
$p\geqslant 5$
or
$p=3$
and
$\mathsf{A}_{7}$
is not involved in
$G$
, and by
$3^{4}$
if
$p=3$
and
$\mathsf{A}_{7}$
is involved in
$G$
.
Let
$C(G)$
be the poset of cyclic subgroups of a finite group
$G$
and let
$\mathscr{P}$
be the class of
$p$
-groups of order
$p^{n}$
(
$n\geq 3$
). Consider the function
$\unicode[STIX]{x1D6FC}:\mathscr{P}\longrightarrow (0,1]$
given by
$\unicode[STIX]{x1D6FC}(G)=|C(G)|/|G|$
. In this paper, we determine the second minimum value of
$\unicode[STIX]{x1D6FC}$
, as well as the corresponding minimum points. Since the problem of finding the second maximum value of
$\unicode[STIX]{x1D6FC}$
has been solved for
$p=2$
, we focus on the case of odd primes in determining the second maximum.
It is well known that the pair
$(\mathcal {S}_n,\mathcal {S}_{n-1})$
is a Gelfand pair where
$\mathcal {S}_n$
is the symmetric group on n elements. In this paper, we prove that if G is a finite group then
$(G\wr \mathcal {S}_n, G\wr \mathcal {S}_{n-1}),$
where
$G\wr \mathcal {S}_n$
is the wreath product of G by
$\mathcal {S}_n,$
is a Gelfand pair if and only if G is abelian.
For a character
$\unicode[STIX]{x1D712}$
of a finite group
$G$
, the co-degree of
$\unicode[STIX]{x1D712}$
is
$\unicode[STIX]{x1D712}^{c}(1)=[G:\text{ker}\unicode[STIX]{x1D712}]/\unicode[STIX]{x1D712}(1)$
. We study finite groups whose co-degrees of nonprincipal (complex) irreducible characters are divisible by a given prime
$p$
.
We define and study cyclotomic quotients of affine Hecke algebras of type B. We establish an isomorphism between direct sums of blocks of these algebras and a generalization, for type B, of cyclotomic quiver Hecke algebras, which are a family of graded algebras closely related to algebras introduced by Varagnolo and Vasserot. Inspired by the work of Brundan and Kleshchev, we first give a family of isomorphisms for the corresponding result in type A which includes their original isomorphism. We then select a particular isomorphism from this family and use it to prove our result.
Let G be a finite group and σ = {σi| i ∈ I} some partition of the set of all primes
$\Bbb{P}$
. Then G is said to be: σ-primary if G is a σi-group for some i; σ-nilpotent if G = G1× … × Gt for some σ-primary groups G1, … , Gt; σ-soluble if every chief factor of G is σ-primary. We use
$G^{{\mathfrak{N}}_{\sigma}}$
to denote the σ-nilpotent residual of G, that is, the intersection of all normal subgroups N of G with σ-nilpotent quotient G/N. If G is σ-soluble, then the σ-nilpotent length (denoted by lσ (G)) of G is the length of the shortest normal chain of G with σ-nilpotent factors. Let Nσ (G) be the intersection of the normalizers of the σ-nilpotent residuals of all subgroups of G, that is,
Then the subgroup Nσ (G) is called the σ-nilpotent norm of G. We study the relationship of the σ-nilpotent length with the σ-nilpotent norm of G. In particular, we prove that the σ-nilpotent length of a σ-soluble group G is at most r (r > 1) if and only if lσ (G/ Nσ (G)) ≤ r.
We enumerate the number of isoclinism classes of semi-extraspecial p-groups with derived subgroup of order p2. To do this, we enumerate GL (2, p)-orbits of sets of irreducible, monic polynomials in 𝔽p[x]. Along the way, we also provide a new construction of an infinite family of semi-extraspecial groups as central quotients of Heisenberg groups over local algebras.
We formulate a
$q$
-Schur algebra associated with an arbitrary
$W$
-invariant finite set
$X_{\text{f}}$
of integral weights for a complex simple Lie algebra with Weyl group
$W$
. We establish a
$q$
-Schur duality between the
$q$
-Schur algebra and Hecke algebra associated with
$W$
. We then realize geometrically the
$q$
-Schur algebra and duality and construct a canonical basis for the
$q$
-Schur algebra with positivity. With suitable choices of
$X_{\text{f}}$
in classical types, we recover the
$q$
-Schur algebras in the literature. Our
$q$
-Schur algebras are closely related to the category
${\mathcal{O}}$
, where the type
$G_{2}$
is studied in detail.
Denote by m(G) the largest size of a minimal generating set of a finite group G. We estimate m(G) in terms of
$\sum _{p\in \pi (G)}d_p(G),$
where we are denoting by dp(G) the minimal number of generators of a Sylow p-subgroup of G and by π(G) the set of prime numbers dividing the order of G.
The class of all monolithic (that is, subdirectly irreducible) groups belonging to a variety generated by a finite nilpotent group can be axiomatised by a finite set of elementary sentences.
Let
$p$
be a prime,
$G$
a solvable group and
$P$
a Sylow
$p$
-subgroup of
$G$
. We prove that
$P$
is normal in
$G$
if and only if
$\unicode[STIX]{x1D711}(1)_{p}^{2}$
divides
$|G:\ker (\unicode[STIX]{x1D711})|_{p}$
for all monomial monolithic irreducible
$p$
-Brauer characters
$\unicode[STIX]{x1D711}$
of
$G$
.
Let
$p$
be an odd prime. We construct a
$p$
-group
$P$
of nilpotency class two, rank seven and exponent
$p$
, such that
$\text{Aut}(P)$
induces
$N_{\text{GL}(7,p)}(G_{2}(p))=Z(\text{GL}(7,p))G_{2}(p)$
on the Frattini quotient
$P/\unicode[STIX]{x1D6F7}(P)$
. The constructed group
$P$
is the smallest
$p$
-group with these properties, having order
$p^{14}$
, and when
$p=3$
our construction gives two nonisomorphic
$p$
-groups. To show that
$P$
satisfies the specified properties, we study the action of
$G_{2}(q)$
on the octonion algebra over
$\mathbb{F}_{q}$
, for each power
$q$
of
$p$
, and explore the reducibility of the exterior square of each irreducible seven-dimensional
$\mathbb{F}_{q}[G_{2}(q)]$
-module.
For a reductive group
$G$
over a finite field, we show that the neutral block of its mixed Hecke category with a fixed monodromy under the torus action is monoidally equivalent to the mixed Hecke category of the corresponding endoscopic group
$H$
with trivial monodromy. We also extend this equivalence to all blocks. We give two applications. One is a relationship between character sheaves on
$G$
with a fixed semisimple parameter and unipotent character sheaves on the endoscopic group
$H$
, after passing to asymptotic versions. The other is a similar relationship between representations of
$G(\mathbb{F}_{q})$
with a fixed semisimple parameter and unipotent representations of
$H(\mathbb{F}_{q})$
.