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1. CMB Online first
$m$-embedded Subgroups and $p$-nilpotency of Finite Groups Let $A$ be a subgroup of a finite group $G$ and $\Sigma : G_0\leq
G_1\leq\cdots \leq G_n$ some subgroup series of $G$. Suppose that
for each pair $(K,H)$ such that $K$ is a maximal subgroup of $H$ and
$G_{i-1}\leq K \lt H\leq G_i$, for some $i$, either $A\cap H = A\cap K$
or $AH = AK$. Then $A$ is said to be $\Sigma$-embedded in $G$; $A$
is said to be $m$-embedded in $G$ if $G$ has a subnormal subgroup
$T$ and a $\{1\leq G\}$-embedded subgroup $C$ in $G$ such that $G =
AT$ and $T\cap A\leq C\leq A$. In this article, some sufficient
conditions for a finite group $G$ to be $p$-nilpotent are given
whenever all subgroups with order $p^{k}$ of a Sylow $p$-subgroup of
$G$ are $m$-embedded for a given positive integer $k$.
Keywords:finite group, $p$-nilpotent group, $m$-embedded subgroup Categories:20D10, 20D15 |
2. CMB 2011 (vol 55 pp. 673)
Multiplicity Free Jacquet Modules Let $F$ be a non-Archimedean local field or a finite field.
Let $n$ be a natural number and $k$ be $1$ or $2$.
Consider $G:=\operatorname{GL}_{n+k}(F)$ and let
$M:=\operatorname{GL}_n(F) \times \operatorname{GL}_k(F)\lt G$ be a maximal Levi subgroup.
Let $U\lt G$ be the corresponding unipotent subgroup and let $P=MU$ be the corresponding parabolic subgroup.
Let $J:=J_M^G: \mathcal{M}(G) \to \mathcal{M}(M)$ be the Jacquet functor, i.e., the functor of coinvariants with respect to $U$.
In this paper we prove that $J$ is a multiplicity free functor, i.e.,
$\dim \operatorname{Hom}_M(J(\pi),\rho)\leq 1$,
for any irreducible representations $\pi$ of $G$ and $\rho$ of $M$.
We adapt the classical method of Gelfand and Kazhdan, which proves the ``multiplicity free" property of certain representations to prove the ``multiplicity free" property of certain functors.
At the end we discuss whether other Jacquet functors are multiplicity free.
Keywords:multiplicity one, Gelfand pair, invariant distribution, finite group Categories:20G05, 20C30, 20C33, 46F10, 47A67 |
3. CMB 2004 (vol 47 pp. 530)
A Characterization of $ PSU_{11}(q)$ Order components of a finite simple group were introduced in [4].
It was proved that some non-abelian simple groups are uniquely determined
by their order components. As the main result of this paper, we
show that groups $PSU_{11}(q)$ are also uniquely determined by
their order components. As corollaries of this result, the
validity of a conjecture of J. G. Thompson and a conjecture of W.
Shi and J. Bi both on $PSU_{11}(q)$ are obtained.
Keywords:Prime graph, order component, finite group,simple group Categories:20D08, 20D05, 20D60 |
4. CMB 2002 (vol 45 pp. 272)
The Transfer in the Invariant Theory of Modular Permutation Representations II In this note we show that the image of the transfer for permutation
representations of finite groups is generated by the transfers of
special monomials. This leads to a description of the image of the
transfer of the alternating groups. We also determine the height of
these ideals.
Keywords:polynomial invariants of finite groups, permutation representation, transfer Category:13A50 |
5. CMB 1999 (vol 42 pp. 125)
Modular Vector Invariants of Cyclic Permutation Representations Vector invariants of finite groups (see the introduction for an
explanation of the terminology) have often been used to illustrate the
difficulties of invariant theory in the modular case: see,
\eg., \cite{Ber}, \cite{norway}, \cite{fossum}, \cite{MmeB},
\cite{poly} and \cite{survey}. It is therefore all the more
surprising that the {\it unpleasant} properties of these invariants
may be derived from two unexpected, and remarkable, {\it nice}
properties: namely for vector permutation invariants of the cyclic
group $\mathbb{Z}/p$ of prime order in characteristic $p$ the
image of the transfer homomorphism $\Tr^{\mathbb{Z}/p} \colon
\mathbb{F}[V] \lra \mathbb{F}[V]^{\mathbb{Z}/p}$ is a prime ideal,
and the quotient algebra $\mathbb{F}[V]^{\mathbb{Z}/p}/ \Im
(\Tr^{\mathbb{Z}/p})$ is a polynomial algebra on the top Chern
classes of the action.
Keywords:polynomial invariants of finite groups Category:13A50 |