Fixed Point Index for G-Equivariant Multivalued Maps

2008

Journal of the American Mathematical Society, 2012

We prove several results on products of conjugacy classes in finite simple groups. The first result is that for any finite nonabelian simple groups, there exists a triple of conjugate elements with product 1 1 which generate the group. This result and other ideas are used to solve a 1966 conjecture of Peter Neumann about the existence of elements in an irreducible linear group with small fixed space. We also show that there always exist two conjugacy classes in a finite nonabelian simple group whose product contains every nontrivial element of the group. We use this to show that every element in a nonabelian finite simple group can be written as a product of two r r th powers for any prime power r r (in particular, a product of two squares answering a conjecture of Larsen, Shalev and Tiep).

International Journal of Group Theory, 2017

Some of the results of this paper are part of the third author's Ph.D. thesis at the University Jaume I of Castellon, who is financially supported by a predoctoral grant of this university. The first and second authors are supported by the Valencian Government, Proyecto PROMETEOII/2015/011. The first and the third authors are also partially supported by Universitat Jaume I, grant P11B2015-77.

European Journal of Pure and Applied Mathematics

A nonempty set G is a g-group [with respect to a binary operation ∗] if it satisfies the following properties: (g1) a ∗ (b ∗ c) = (a ∗ b) ∗ c for all a, b, c ∈ G; (g2) for each a ∈ G, there exists an element e ∈ G such that a ∗ e = a = e ∗ a (e is called an identity element of a); and, (g3) for each a ∈ G, there exists an element b ∈ G such that a ∗ b = e = b ∗ a for some identity element eof a. In this study, we gave some important properties of g-subgroups, homomorphism of g-groups, andthe zero element. We also presented a couple of ways to construct g-groups and g-subgroups.

Comptes Rendus. Mathématique, 2020

A group is nested if the centers of the irreducible characters form a chain. In this paper, we will show that there is a set of subgroups associated with the conjugacy classes of group so that a group is nested if and only if these subgroups form a chain.

Mathematics and Statistics, 2014

In this paper, we have defined the concept of p-map and studied some properties of p-map. By using this map, we have shown that p(G) is a subgroup of G and S = {x : p(x) = e} is a right transversal (with identity) of p(G) in G which becomes group by using p-map and some more conditions. Finally, we have shown that G be an extension of p(G).

Proceedings of the American Mathematical Society, 1983

Let G be a finite group. In this note we study the question of realizing a collection of graded commutative algebras over Q as the cohomology algebras with rational coefficients of the fixed point sets X" ( H < G) of a G-space X. Let tí be a graded commutative algebra over Q. The question of realizing & as the cohomology with rational coefficients of a space X is answered by Quillen [4] and more directly by Sullivan [5], In particular, Sullivan constructs a space X of finite type, i.e. ir¡(X) is a finitely generated abelian group for every /, which realizes 68. Now let G be a finite group which acts on tf from the left by algebra isomorphisms. Because of the functoriality of the constructions in and one can construct a G-space X such that H*(X;Q) = tf, where the isomorphism is G-equivariant. The space X in both cases is a rational space. In this note we consider a more general question. Let 6C be the category of canonical orbits of a finite group G [1], The objects of 0C are the quotient spaces G/H, where H is a subgroup of G (H < G), and the morphisms are the G-maps between them, where G acts on G/H by left multiplication. Definition 1. A system of graded commutative algebras (GA's) for G is a covariant functor from (3C into the category of graded commutative connected algebras over Q. We recall that a GA tí is said to be connected if 68° = Q and is said to be of finite type if tf " is a finite-dimensional vector space over Q. Let A be a G-space such that each fixed point set XH, H < G, is nonempty and connected. Given A, a system of GA's H*( X) is defined by H*(X)(G/H) =H*(X";Q) on objects of (P0. If/: G/H -G/K is a G-map, then there exists an element g E G such that g~lHg < K, and the map /is determined by H t-> gK. The map/induces a map /: XK -» X" by x i-> gx and therefore a unique map H*(X)(f)=f*: H*(X";Q) -H*(XK;Q). The main result of this paper is the following Theorem 2. Given a system H of connected GA's of finite type, there exists a G-CW-complex X of finite type such that H*(X) = H.

Communications in Algebra, 2014

The work is inspired by an article of M. Herzog, P. Longobardi, and M. Maj, who considered groups with a nite number of innite conjugacy classes. Their main results were obtained under assumption that the F C-center is of nite index in the group. We consider here innite groups with a nite number of conjugacy classes of any size (F N CC-groups). Hence the F C-center in our case will be nite, but of innite index in the group. Among results on these groups we give a criterion for a wreath product of F N CC-groups to be an F N CC-group.

We implement GAP functions about groups with action on itself and investigate some basic properties of small groups with action on itself of order $<32$.

Journal of Pure and Applied Algebra, 1994

Communications in Algebra, 2009

Algebra and Logic, 1980

In the recent past a series of strong results have been announced, which essentially constitute an exhaustive treatment of the problem of describing the p -local structure of finite groups p of type characteristic two in the case where the G -rank ( p an odd prime), of the 2-local subgroups of G is sufficiently big (viz°, ~ ). The situation is much less clear in the case of small fl -rank. Here, it seems that a characterization would be useful of known simple groups, not necessarily of type characteristic two, by means of the centralizers of elements of order /D, or -in the first place -by means of the centralizers of elements of order three.

Topology and its Applications, 2004

Let G be a finite group acting freely in a Hausdorff, paracompact, connected and locally pathwise connected topological space X such that H i (X, Z) = 0 for 0 < i < m and H m+1 (G, Z) = 0. Let f : X → Y be a map of X to a finite k-dimensional CW-complex Y. We show that if m ≥ |G|k, then f has a (H, G)coincidence point for some nontrivial subgroup H of G.

2010

This thesis deals with the conjugacy problem in groups and its twisted variants. We analyze recent results by Bogopolski, Martino, Maslakova and Ventura on the twisted conjugacy problem in free groups and its implication for the conjugacy problem in free-by-cyclic groups and some further group extensions. We also consider the doubly-twisted conjugacy problem in free groups. Staecker has developed an algorithm for deciding doubly-twisted conjugacy relations in the case where the involved homomorphisms satisfy