Some properties of abnormal subgroups in generalized soluble groups will be considered. In particular, the transitivity of abnormality in metahypercentral groups is proven. Also it will be proven that a subgroup H of a radical group G is abnormal in G if and only if every intermediate subgroup for H coincides with its normalizer in G. This result will extend on radical groups the well-known criterion of abnormality for finite soluble groups obtained by D. Taunt. For some infinite groups (not only periodic) the existence of Carter subgroups and their conjugations will be also proven.
In the paper it is proven that Carter subgroups of a finite group are conjugate. A complete classification of Carter subgroups in finite almost simple groups is also obtained.
It is proven in the paper, that Carter subgroups of a finite group are conjugate if Carter subgroups in the group of induced automorphisms for every non-Abelian composition factor are conjugate.
In the paper we complete the classification of Carter subgroups in finite almost simple groups. In particular, we prove that Carter subgroups of every finite almost simple group are conjugate. Togeather with previous results by author and F. Dalla Volta, A. Lucchini, and M. C. Tamburini, as a corollary, it follows that Carter subgroups of every finite group are conjugate.
In the note we prove that all composition factors of a finite group possessing a Carter subgroup of odd order either are abelain, or are isomorphic to $L_2(3^{2n+1})$.
Let $π$ be a set of primes. We show that $π$-separable groups have a conjugacy class of subgroups which specialize to Carter subgroups, i.e. self-normalizing nilpotent subgroups, or equivalently, nilpotent projectors, when specializing to soluble groups.
In the paper we obtain the existence criterion of a Carter subgroup in a finite group in terms of its normal series. An example showing that the criterion cannot be reformulated in terms of composition factors is given.
We consider an amalgam of groups constructed from fusion systems for different odd primes p and q. This amalgam contains a self-normalizing cyclic subgroup of order pq and isolated elements of order p and q.
Let $G$ be a finite solvable group and $H$ be a subgroup of $Aut(G)$. Suppose that there exists an $H$-invariant Carter subgroup $F$ of $G$ such that the semidirect product $FH$ is a Frobenius group with kernel $F$. We prove that the terms of the Fitting series of $C_{G}(H)$ are obtained as the intersection of $C_{G}(H)$ with the corresponding terms of the Fitting series of $G$, and the Fitting height of $G$ may exceed the Fitting height of $C_{G}(H)$ by at most one. As a corollary it is shown that for any set of primes $π$, the terms of the $π$-series of $C_{G}(H)$ is obtained as the intersec…
We modify the Frecon-Jaligot construction of Carter subgroups to show that a degenerate type group has a Carter subgroup invariant under the Sylow 2-subgroup of a group of automorphisms; thus reducing the need to know that Carter subgroups are conjugate in degenerate type groups.
This paper provides new, relatively simple proofs of some important results about unipotent classes in simple linear algebraic groups. We derive the formula for the Jordan blocks of the Richardson class of a parabolic subgroup of a classical group. This result was originally due to Spaltenstein. Secondly, we derive, for good characteristic, the description of the natural partial order of unipotent classes of a classical group in terms of their Jordan blocks. This result was originally due to Gerstenhaber and Hesselink. As a consequence we obtain a proof of the Bala--Carter Theorem which holds …
This paper describes how to use subgroups to parameterize unipotent classes in the classical algebraic group in characteristic 2. These results can be viewed as an extension of the Bala-Carter Theorem, and give a convenient way to compare unipotent classes in a group $G$ with unipotent classes of a subgroup $X$ where $G$ is exceptional and $X$ is a Levi subgroup of classical type.
We present unpublished work of D.Carter, G.Keller, and E.Paige on bounded generation in special linear groups. Let n be a positive integer, and let A = O be the ring of integers of an algebraic number field K (or, more generally, let A be a localization O_S.) If n = 2, assume that A has infinitely many units. We show there is a finite-index subgroup H of SL(n,A), such that every matrix in H is a product of a bounded number of elementary matrices. We also show that if T is in SL(n,A), and T is not a scalar matrix, then there is a finite-index, normal subgroup N of SL(n,A), such that every eleme…
A finite group $P$ is said to be \emph{primary} if $|P|=p^{a}$ for some prime $p$. We say a primary subgroup $P$ of a finite group $G$ satisfies the \emph{Frobenius normalizer condition} in $G$ if $N_{G}(P)/C_{G}(P)$ is a $p$-group provided $P$ is $p$-group. In this paper, we determine the structure of a finite group $G$ in which every non-subnormal primary subgroup satisfies the Frobenius normalized condition. In particular, we prove that if every non-normal primary subgroup of $G$ satisfies the Frobenius condition, then $G/F(G)$ is cyclic and every maximal non-normal nilpotent subgroup $U$ o…
We consider factorizations of a finite group $G$ into conjugate subgroups, $G=A^{x_{1}}\cdots A^{x_{k}}$ for $A\leq G$ and $x_{1},\ldots ,x_{k}\in G$, where $A$ is nilpotent or solvable. First we exploit the split $BN$-pair structure of finite simple groups of Lie type to give a unified self-contained proof that every such group is a product of four or three unipotent Sylow subgroups. Then we derive an upper bound on the minimal length of a solvable conjugate factorization of a general finite group. Finally, using conjugate factorizations of a general finite solvable group by any of its Carter…
A proper subgroup $H$ of a group $G$ is said to be: $\Bbb{P}$-subnormal in $G$ if there exists a chain of subgroups $H=H_0 < H_1< ... < H_{n}=G$ such that $|H_{i}:H_{i-1}|$ is a prime for $i=1,...,n$; $\Bbb{P}$-abnormal in $G$ if for every two subgroups $K\leq L$ of $G$, where $H\leq K$, $|L:K|$ is not a prime. In this paper we describe finite groups in which every non-identity subgroup is either $\Bbb{P}$-subnormal or $\Bbb{P}$-abnormal.
One of the central tools in the classification of simple algebraic groups is the distinction between semisimple subgroups and unipotent subgroups. It is not a priori clear how to make this distinction for torsion-free subgroups of a group of finite Morley rank. We exploit the ``graded'' notion of 0-unipotence to develop a Sylow theory for torsion-free subgroups of a solvable group of finite Morley rank. This Sylow theory provides a robust alternative to the usual theory of Carter subgroups, and will be used in the analysis of intersections of Borel subgroups in minimal simple groups.
In this article, we study group theoretical embedding properties of subgroups in central products of finite groups. Specifically, we give characterizations of normal, subnormal, and abnormal subgroups of a central product of two groups.
Università degli Studi di Modena e Reggio Emilia（摩德纳和雷焦艾米利亚大学）；University of Southern California（南加州大学）；University of Lincoln（林肯大学）；University of Brasilia（巴西利亚大学）
