Nick Gill, László Pyber, Ian Short, Endre Szabó
We prove that if $G$ is a finite simple group of Lie type and $S$ a subset of $G$ of size at least two then $G$ is a product of at most $c\log|G|/\log|S|$ conjugates of $S$, where $c$ depends only on the Lie rank of $G$. This confirms a conjecture of Liebeck, Nikolov and Shalev in the case of families of simple groups of Lie type of bounded rank.
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We propose a general conjecture on decompositions of finite simple groups as products of conjugates of an arbitrary subset. We prove this conjecture for bounded subsets of arbitrary finite simple groups, and for large subsets of groups of Lie type of bounded rank. Some of our arguments apply recent advances in the theory of growth in finite simple groups of Lie type, and provide a variety of new product decompositions of these groups.
In this paper we use JSJ-decompositions to formalise a folk conjecture recorded by Pride on the structure of one-relator groups with torsion. We prove a slightly weaker version of the conjecture, which implies that the structure of one-relator groups with torsion closely resemble the structure of torsion-free hyperbolic groups.
The Alperin weight conjecture was reduced to simple groups by the work of Navarro, Tiep and Späth. To prove Alperin weight conjecture, it suffices to show that all finite non-abelian simple groups are BAW-good. We reduce the verification of the inductive conditons for groups of Lie type in non-defining characteristic to quasi-isolated blocks.
I. Hambleton, L. Taylor and B. Williams conjectured a general formula in spirit of H. Lenstra for the decomposition of $G_n(RG)$ for any finite group $G$ and noetherian ring $R.$ The conjectured decomposition was shown to hold for some large classes of finite groups. D. Webb and D. Yao discovered that the conjecture failed for the symmetric group $S_5$, but remarked that it still might be reasonable to expect the HTW-decomposition for solvable groups. In this paper we show that the solvable group $\mathrm{SL}(2,\mathbb{F}_3)$ is also a counterexample to the conjectured HTW-decomposition. Furth…
Nonuniqueness of semidirect decompositions of groups is an insufficiently studied question in contrast to direct decompositions. We obtain some results about semidirect decompositions for semidirect products with factors which are nontrivial direct products. We deal with a special case of semidirect product when the twisting homomorphism acts diagonally on a direct product, as well as for the case when the extending group is a direct product. We give applications of these results in the case of generalized dihedral groups and classic dihedral groups $D_{2n}$. For $D_{2n}$ we give a complete de…
Group Embedding into Wreath Products: Cartesian Decomposition Approach
本文利用笛卡尔分解，将正则群和对角群嵌入到乘积作用下的圈积中，提出了一种新的嵌入方法。
For a subgroup of a free product of finite groups, we obtain necessary conditions (on its Kurosh decomposition) to be verbally closed.
We prove that every quasisimple group of classical type is a product of boundedly many conjugates of a quasisimple subgroup of type A_n.
A polynomial-time algorithm is produced which, given generators for a group of permutations on a finite set, returns a direct product decomposition of the group into directly indecomposable subgroups. The process uses bilinear maps and commutative rings to characterize direct products of p-groups of class 2 and reduces general groups to p-groups using group varieties. The methods apply to quotients of permutation groups and operator groups as well.
We classify all pairs (G,V) with G a closed subgroup in a classical group with natural module V over the complex numbers such that G has the same composition factors on the kth tensor power of V, for a fixed (small) k. In particular, we prove Larsen's conjecture stating that for dim(V) > 6 and k = 4, there are no such G aside from those containing the derived subgroup of the classical group. We also find all the examples where this fails for dim(V) < 7. As a consequence of our results, we obtain a short proof of a related conjecture of Katz. These conjectures are used in Katz's recent works on…
Let $G$ be a finite group and $N$ be a normal subgroup of $G$. Let $J=J(F[N])$ denote the Jacboson radical of $F[N]$ and $I={\rm Ann}(J)=\{α\in F[G]|Jα=0\}$. We have another algebra $F[G]/I$. We study the decomposition of Cartan matrix of $F[G]$ according to $F[G/N]$ and $F[G]/I$. This decomposition establishs some connections between Cartan invariants and chief composition factors of $G$. We find that existing zero-defect $p$-block in $N$ depends on the properties of $I$ in $F[G]$ or Cartan invariants. When we consider the Cartan invariants for a block algebra $B$ of $G$, the decomposition is…
We show that a linear decomposition attack based on the decomposition method introduced by the author works by finding the exchanged secret keys in all main protocols using semidirect products of (semi)grops proposed by Kahrobaei, Shpilrain, Habeeb, Koupparis and Lam.
In this paper we introduce a new family of icc groups $Γ$ which satisfy the following product rigidity phenomenon, discovered in [DHI16] (see also [dSP17]): all tensor product decompositions of the II$_1$ factor $L(Γ)$ arise only from the canonical direct product decompositions of the underlying group $Γ$. Our groups are assembled from certain HNN-extensions and amalgamated free products and include many remarkable groups studied throughout mathematics such as graph product groups, poly-amalgam groups, Burger-Mozes groups, Higman group, various integral two-dimensional Cremona groups, etc. As …
Let ${\mathbf G}^F$ be a finite group of Lie type, where ${\mathbf G}$ is a reductive group defined over ${\overline{\mathbb F}_q}$ and $F$ is a Frobenius root. Lusztig's Jordan decomposition parametrises the irreducible characters in a rational series${\mathcal E}({{\mathbf G}^F},(s)_{{\mathbf G}^{*F^*}})$ where $s\in{{\mathbf G}^{*F^*}}$ by the series ${\mathcal E}(C_{{\mathbf G}^*}(s)^{F^*},1)$.We conjecture that the Shintani twisting preserves the space of class functions generated by the union of the ${\mathcal E}({{\mathbf G}^F},(s')_{{\mathbf G}^{*F^*}})$ where$(s')_{{\mathbf G}^{*F^*}}…
Free Pro-2 Product Decompositions and Hilbert's Theorem 90 for the Kaplansky Radical
本文针对有限平方类域建立了确保Kaplansky根的Hilbert第90型性质的伽罗瓦理论判据，将其应用于初等类型极大pro-2伽罗瓦群的域等，为有限维H猜想的已知正例提供了统一解释。
We use tilting modules to study the structure of the tensor product of two simple modules for the algebraic group $\SL_2$, in positive characteristic, obtaining a twisted tensor product theorem for its indecomposable direct summands. Various other related results are obtained, and numerous examples are computed.
We show that a tensor product of irreducible, finite dimensional representations of a simple Lie algebra over a field of characteristic zero, determines the individual constituents uniquely. This is analogous to the uniqueness of prime factorisation of natural numbers.
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