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Which finite simple groups are unit groups?  [PDF]
Christopher Davis,Tommy Occhipinti
Mathematics , 2014, DOI: 10.1016/j.jpaa.2013.08.013
Abstract: We prove that if $G$ is a finite simple group which is the unit group of a ring, then $G$ is isomorphic to either (a) a cyclic group of order 2; (b) a cyclic group of prime order $2^k -1$ for some $k$; or (c) a projective special linear group $PSL_n(\mathbb{F}_2)$ for some $n \geq 3$. Moreover, these groups do (trivially) all occur as unit groups. We deduce this classification from a more general result, which holds for groups $G$ with no non-trivial normal 2-subgroup.
Extension of p-local finite groups  [PDF]
Carles Broto,Natalia Castellana,Jesper Grodal,Ran Levi,Bob Oliver
Mathematics , 2005,
Abstract: A p-local finite group consists of a finite p-group S, together with a pair of categories which encode ``conjugacy'' relations among subgroups of S, and which are modelled on the fusion in a Sylow p-subgroup of a finite group. It contains enough information to define a classifying space which has many of the same properties as p-completed classifying spaces of finite groups. In this paper, we study and classify extensions of p-local finite groups, and also compute the fundamental group of the classifying space of a p-local finite group.
Beauville structures in finite p-groups  [PDF]
Gustavo A. Fernández-Alcober,?ükran Gül
Mathematics , 2015,
Abstract: We study the existence of (unmixed) Beauville structures in finite p-groups. First of all, we extend Catanese's characterisation of abelian Beauville groups to finite p-groups satisfying a certain condition which is much weaker than commutativity. Our result applies in particular to regular p-groups, powerful p-groups and p-central p-groups, and thus also to all p-groups of order at most p^p. On the other hand, we determine which quotients of the Nottingham group over F_p are Beauville groups, for an odd prime p. As a consequence, we give the first explicit infinite family of Beauville 3-groups.
Finite morphic $p$-groups  [PDF]
A. Caranti,C. M. Scoppola
Mathematics , 2014,
Abstract: According to Li, Nicholson and Zan, a group $G$ is said to be morphic if, for every pair $N_{1}, N_{2}$ of normal subgroups, each of the conditions $G/N_{1} \cong N_{2}$ and $G/N_{2} \cong N_{1}$ implies the other. Finite, homocyclic $p$-groups are morphic, and so is the nonabelian group of order $p^{3}$ and exponent $p$, for $p$ an odd prime. It follows from results of An, Ding and Zhan on self dual groups that these are the only examples of finite, morphic $p$-groups. In this paper we obtain the same result under a weaker hypotesis.
Groups which are almost groups of Lie type in characteristic p  [PDF]
Chris Parker,Gernot Stroth
Mathematics , 2011,
Abstract: For a prime $p$, a $p$-subgroup of a finite group $G$ is said to be large if and only if $Q= F^*(N_G(Q))$ and, for all $1 \neq U \le Z(Q)$, $N_G(U) \le N_G(Q)$. In this article we determine those groups $G$ which have a large subgroup and which in addition have a proper subgroup $H$ containing a Sylow $p$-subgroup of $G$ with $F^*(H)$ a group of Lie type in characteristic $p$ and rank at least 2 (excluding $\PSL_3(p^a)$) and $C_H(z)$ soluble for some $z \in Z(S)$. This work is part of a project to determine the groups $G$ which contain a large $p$-subgroup.
A Survey on Automorphism Groups of Finite $p$-Groups  [PDF]
Geir T. Helleloid
Mathematics , 2006,
Abstract: This survey on the automorphism groups of finite p-groups focuses on three major topics: explicit computations for familiar finite p-groups, such as the extraspecial p-groups and Sylow p-subgroups of Chevalley groups; constructing p-groups with specified automorphism groups; and the discovery of finite p-groups whose automorphism groups are or are not p-groups themselves. The material is presented with varying levels of detail, with some of the examples given in complete detail.
Class Preserving Automorphisms of Finite p-groups  [PDF]
Manoj K. Yadav
Mathematics , 2005, DOI: 10.1112/jlms/jdm025
Abstract: We classify all finite $p$-groups $G$ for which |$Aut_{c}(G)$| attains its maximum value, where $Aut_{c}(G)$ denotes the group of all class preserving automorphisms of $G$ .
Nilpotent $p$-local finite groups  [PDF]
J. Cantarero,J. Scherer,A. Viruel
Mathematics , 2011, DOI: 10.1007/s11512-013-0181-4
Abstract: In this paper we prove characterizations of $p$-nilpotency for fusion systems and $p$-local finite groups that are inspired by results in the literature for finite groups. In particular, we generalize criteria by Atiyah, Brunetti, Frobenius, Quillen, Stammbach and Tate.
Gray isometries for finite $p$-groups
Reza Sobhani
Transactions on Combinatorics , 2013,
Abstract: We construct two classes of Gray maps, called type-I Gray map and type-II Gray map, for a finite $p$-group $G$. Type-I Gray maps are constructed based on the existence of a Gray map for a maximal subgroup $H$ of $G$. When $G$ is a semidirect product of two finite $p$-groups $H$ and $K$, both $H$ and $K$ admit Gray maps and the corresponding homomorphism $psi:Hlongrightarrow {rm Aut}(K)$ is compatible with the Gray map of $K$ in a sense which we will explain, we construct type-II Gray maps for $G$. Finally, we consider group codes over the dihedral group $D_8$ of order 8 given by the set of their generators, and derive a representation and an encoding procedure for such codes.
The Roquette category of finite p-groups  [PDF]
Serge Bouc
Mathematics , 2011,
Abstract: Let p be a prime number. This paper introduces the Roquette category R_p of finite p-groups, which is an additive tensor category containing all finite p-groups among its objects. In R_p, every finite p-group P admits a canonical direct summand, called the edge of P. Moreover P splits uniquely as a direct sum of edges of Roquette p-groups, and the tensor structure of R_p can be described in terms of such edges. The main motivation for considering this category is that the additive functors from R_p to abelian groups are exactly the rational p-biset functors. This yields in particular very efficient ways of computing such functors on arbitrary p-groups : this applies to the representation functors R_K, where K is any field of characteristic 0, but also to the functor of units of Burnside rings, or to the torsion part of the Dade group.
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