Conjugacy Classes in Sylow p-Subgroups of Finite Chevalley Groups in Bad Characteristic

Let U = U(q) be a Sylow p-subgroup of a finite Chevalley group G = G(q). Röhrle and Goodwin in 2009 determined a parameterization of the conjugacy classes of U, for G of small rank when q is a power of a good prime for G. As a consequence they verified that the number k(U) of conjugacy classes of U is given by a polynomial in q with integer coefficients. In the present paper, we consider the case when p is a bad prime for G. Our motivation is to observe how the situation differs between good and bad characteristics. We obtain a parameterization of the conjugacy classes of U, when G has rank less than or equal to 4, and G is not of type F ₄. In these cases we deduce that k(U) is given by a polynomial in q with integer coefficients; this polynomial is different from the polynomial for good primes.     

On the regular Sylow p-subgroups of Chevalley groups over \mathbb{Z}_{p^m }

We prove that a Sylow p-subgroup of the general linear group of dimension n over the residue ring modulo p ^m is regular for n ² < p and powerful if and only if n = 2 and m = 1. We obtain similar results for the Sylow p-subgroups of normal types over the same ring.     

有限交换环上典型群的Sylow子群

令Ｒ为有限交换局部环，Ｍ表其唯一的极大理想，ｋ表其剩余类域．本文定出了Ｒ上的一般线性群ＧＬｎＲ，辛群Ｓｐ２ｎＲ及双曲正交群Ｏ２ｎＲ的Ｓｙｌｏｗ子群．一般讲，若ｃｈａｒｘ＝ｐ，上述三类典型群的Ｓｙｌｏｗ ｐ－子群分别同构于由某些特殊形式的矩阵生成的子群；若ｃｈａｒｋ≠ｐ，上述三类典型群的Ｓｙｌｏｗｐ－子群分别同构于一循环群或半二面体群与若干Ｚｐ型循环群的圈积。     

The generic character table of a Sylow p-subgroup of a finite Chevalley group of type D4

Let U be a Sylow p-subgroup of the finite Chevalley group of type D₄ over the field of q elements, where q is a power of a prime p. We describe a construction of the generic character table of U.     

Standardness and Standard Automorphisms of Chevalley Groups, I: the Case of Rank at Least Two

The author studies the linkage between the standardness and the standard automorphisms of Chevalley groups over rings.It is proved that if H is any standard subgroup of G(R),then each of its automorphisms can be extended to an automorphism of G(R,I),restricted to an automorphism of E(R,I),and an automorphism of E(R,I) can be extended to one of G(R,I).The case of Chevalley groups of rank at least two is treated in this paper.Further results about the case of Chevalley groups of rank one,the case of non-commutative ground ring and some others exceptions will appear elsewhere.     

Hypercentral Series and Paired Intersections of Sylow Subgroups of Chevalley Groups

Let G(k) be the Chevalley group of normal type associated with a root system G = , or of twisted type G = ^m,m = 2,3, over a field K. Its root subgroups X_s, for all possible s G⁺, generate a maximal unipotent subgroup U = UG(k) if p = charK < 0, U is a Sylow p-subgroup of G(K). We examine G and K for which there exists a paired intersection U U⁹, g G(K), which is not conjugate in G(K) to a normal subgroup of U. If K is a finite field, this is equivalent to a condition that the normalizer of U U⁹ in G(K)has a p-multiple index. Put p() = max(r,r)/(s,s) | r,s . We prove a statement (Theorem 1) saying the following. Let G(K) be a Chevalley group of Lie rank greater than 1 over a finite field K of characteristic p and U be its Sylow p-subgroup equal to UG(K); also, either G = and p() is distinct from p and 1, or G(K) is a twisted group. Then G(K) contains a monomial element n such that the normalizer U of Uⁿ in G(K) has a p-multiple index. Let K be an associative commutative ring with unity and (K,J) be a congruence subgroup of the Chevalley group (K) modulo a nilpotent ideal J. We examine an hypercentral series 1 Z₁ Z₂ ... Z_c-1 of the group U(K) (K,J). Theorem 2 shows that under an extra restriction on the quotient (J_t : J) of ideals, central series are related via Z_i = T_c-iC, 1 i < c, where C is a subgroup of central diagonal elements. Such a connection exists, in particular, if K = Z_pm and J = (p^d), 1 d < m, d| m.     

The Sylow p-Subgroups of a Limiting (Stable) Linear Group over an Algebraically Closed Field

In Math. Zh. 36(3) (1963), 240–246, we considered the classification of Sylow p-subgroups of a countable (finitary) symmetric group. Later, we extended this classification to Sylow p-subgroups of a limiting (stable) linear group over a finite field GF(q) with char(GF(q)) p. In this paper, the classification of the Sylow p-subgroups of the limiting (stable) linear group GL() is given for an algebraically closed field of characteristic p. Every Sylow p-subgroup is characterized by a p-adic integer together with a finite or countable cardinal.     

有限域上型$B_n$的Chevalley群之间的同态

研究了特征为$p$的有限域上型$B_{n}$的Chevalley群的结构,并确定了特征为$p \ (p\neq 2)$的有限域上型$B_{n}$的Chevalley群 之间的非平凡同态.     

On sylowizers in finite <Emphasis Type="Italic">p</Emphasis>-soluble groups

In general, given a finite group G, a prime p and a p-subgroup R of G, the sylowizers of R in G are not conjugate. In this paper we afford some conditions to achieve the conjugation of the sylowizers of R in a p-soluble group G, among others

Rationality and strong reality for Sylow 2-subgroups of Weyl and alternating groups

It is proved that values of all complex characters for a Sylow 2-subgroup P of any Weyl group are rational, and every element of P is a product of two involutions in P. Similar results hold also for Sylow 2-subgroups of alternating groups.Supported by RFBR grant No. 03-01-00905.Translated from Algebra i Logika, Vol. 44, No. 1, pp. 44–53, January–February, 2005.     

On the existence of solvable normal subgroups in finite groups

The structure of a finite group in dependence on the structure of the subgroups generated by elements of its conjugate class is considered. Translated fromMatematischeskie Zametki, Vol. 61, No. 5, pp. 755–758, May, 1997. Translated by A. I. Shtern     

Finite groups with seminormal Sylow subgroups

In this paper, we prove the following theorem： Let p be a prime number, P a Sylow psubgroup of a group G and π = π（G） ／ {p}. If P is seminormal in G, then the following statements hold： 1） G is a p-soluble group and P＇ ≤ Op（G）; 2） lp（G） ≤ 2 and lπ（G） ≤ 2; 3） if a π-Hall subgroup of G is q-supersoluble for some q ∈ π, then G is q-supersoluble.     

On s-Semipermutable Maximal and Minimal Subgroups of Sylow p-Subgroups of Finite Groups

Let R be a Noetherian algebra over a field k. A formula is given for the Krull dimension of the ring R?_k k(X) in terms of the heights of simple modules with large endomorphism rings.     

具有一个T．I．Sylow 2-子群的有限群的类保持Coleman自同构

设G是一个有限群,它的Sylow 2-子群是T.I.集,证明了如果G的2的方幂阶类保持自同构在G任意的Sylow子群上的限制等于G的某个内自同构的限制,则它一定是一个内自同构.对这样的自同构的研究是由整群环的同构问题所引起的.     

Groups of Automorphisms of Tournaments

By the Holland Representation Theorem, every lattice ordered group (l-group) is isomorphic to a subalgebra of the l-group of automorphisms of a chain. Since weakly associative lattice groups (wal-groups) and tournaments are non-transitive generalizations of l-groups and chains, respectively, the problem concerning the possibility of representation of wal-groups by automorphisms of tournaments arises. In the paper we describe the class of wal-groups isomorphic to wal-groups of automorphisms of tournament and show some of its properties.     

Finite groups in which the normalizers of Sylow 3-subgroups are of odd or primary index

We determine the composition factors of the finite groups in which the normalizers of Sylow 3-subgroups are of odd or primary index.     

Isomorphisms of adjoint Chevalley groups over integral domains

It is shown that every automorphism of an adjoint Chevalley group over an integral domain containing the rational number field is uniquely expressible as the product of a ring automorphism, a graph automorphism and an inner automorphism while every isomorphism between simple adjoint Chevalley groups can be expressed uniquely as the product of a ring isomorphism, a graph isomorphism and an inner automorphism. The isomorphisms between the elementary subgroups are also found having analogous expressions.