{Using smooth one-fixed-point actions on spheres and a result due to Bob Oliver on the tangent representations at fixed points for smooth group actions on disks, we obtain a similar result for ...perfect group actions on spheres. For a finite group G, we compute a certain subgroup IO'(G) of the representation ring RO(G). This allows us to prove that a finite perfect group G has a smooth 2--proper action on a sphere with isolated fixed points at which the tangent representations of G are mutually nonisomorphic if and only if G contains two or more real conjugacy classes of elements not of prime power order. Moreover, by reducing group theoretical computations to number theory, for an integer n \ge 1 and primes p, q, we prove similar results for the group G = A_{n}, \operatorname{SL} _{2}(\mathbb{F} _{p}), or {\operatorname{PSL}} _{2}(\mathbb{F} _{q}). In particular, G has Smith equivalent representations that are not isomorphic if and only if n \ge 8, p \ge 5, q \ge 19.}
We introduce new classes of compact metric spaces: Cannon—Štan'ko, Cainian, and nonabelian compacta. In particular, we investigate compacta of cohomological dimension one with respect to certain ...classes of nonabelian groups, e.g., perfect groups. We also present a new method of constructing compacta with certain extension properties.
Provider: Czech digital library/Česká digitální knihovna - Institution: Academy of Sciences Library/Knihovna Akademie věd ČR - Data provided by Europeana Collections- Ruifang Chen, Xianhe Zhao.- ...Obsahuje bibliografické odkazy- Let $G$ be a finite group. A normal subgroup $N$ of $G$ is a union of several $G$-conjugacy classes, and it is called $n$-decomposable in $G$ if it is a union of $n$ distinct $G$-conjugacy classes. In this paper, we first classify finite non-perfect groups satisfying the condition that the numbers of conjugacy classes contained in its non-trivial normal subgroups are two consecutive positive integers, and we later prove that there is no non-perfect group such that the numbers of conjugacy classes contained in its non-trivial normal subgroups are 2, 3, 4 and 5.- All metadata published by Europeana are available free of restriction under the Creative Commons CC0 1.0 Universal Public Domain Dedication. However, Europeana requests that you actively acknowledge and give attribution to all metadata sources including Europeana
Let M3be a 3-manifold containing no 2-sided projective plane. Let G ≠ {1} be a finitely-generated subgroup of π1(M3) such that G is indecomposable relative to free product, and such that G ...abelianized is finite. (G is "practically perfect".) Then, it is shown that there is a compact 3-submanifold$Z^3 \subset M^3$such that π1(Z3) contains a subgroup of finite index conjugate to G, and Z3is bounded by a 2-sphere. Some related extensions of this result are given, plus an application to compact absolute neighborhood retracts in 3-manifolds.
Perfect Prishchepov groups Chinyere, Ihechukwu; Bainson, Bernard Oduoku
Journal of algebra,
12/2021, Letnik:
588
Journal Article
Recenzirano
Odprti dostop
We study cyclically presented groups of type F to determine when they are perfect. It turns out that to do so, it is enough to consider the Prishchepov groups, so modulo a certain conjecture, we ...classify the perfect Prishchepov groups P(r,n,k,s,q) in terms of the defining integer parameters r,n,k,s,q. In particular, we obtain a classification of the perfect Campbell and Robertson's Fibonacci-type groups H(r,n,s), thereby proving a conjecture of Williams, and yielding a complete classification of the groups H(r,n,s) that are connected Labelled Oriented Graph groups.
Groups that can be approximated by finite groups have been the subject of extensive research. This has led to the investigations of the subgroups of algebraic ultraproducts of finite groups, i.e., ...LEF groups. This paper addresses the dual problem: what are the abstract quotients of ultraproducts of finite groups? Certain cases were already well-studied. For example, if we have an ultraproduct of solvable group, then it is well-known that any finite abstract quotients must still be solvable.
This paper studies the case of ultraproducts of finite perfect groups. We shall show that any finite group could be a quotient of an ultraproduct of finite perfect groups. We provide an explicit construction on how to achieve this for each finite group.
Schur homotopy of the simplest group Giffen, Charles H.
Bulletin (new series) of the American Mathematical Society,
01/1974, Letnik:
80, Številka:
5
Journal Article
Let G be the group of all Interval Exchange Transformations. Results of Arnoux-Fathi (Arn81b), Sah (Sah81) and Vorobets (Vor17) state that G 0 the subgroup of G generated by its commutators is ...simple. In (Arn81b), Arnoux proved that the group G of all Interval Exchange Transformations with flips is simple. We establish that every element of G has a commutator length not exceeding 6. Moreover, we give conditions on G that guarantee that the commutator lengths of the elements of G 0 are uniformly bounded, and in this case for any g ∈ G 0 this length is at most 5. As analogous arguments work for the involution length in G, we add an appendix whose purpose is to prove that every element of G has an involution length not exceeding 12.
We describe the groups that have the same holomorph as a finite perfect group. Our results are complete for centerless groups.
When the center is non-trivial, some questions remain open. The ...peculiarities of the general case are illustrated by a couple of examples that might be of independent interest.
Characters of \pi'-degree Eugenio Giannelli; A. A. Schaeffer Fry; Carolina Vallejo Rodríguez
Proceedings of the American Mathematical Society,
11/2019, Letnik:
147, Številka:
11
Journal Article
Recenzirano
Odprti dostop
Let G be a finite group and let \pi be a set of primes. Write \operatorname {Irr}_{\pi '}(G) for the set of irreducible characters of degree not divisible by any prime in \pi . We show that if \pi ...contains at most two prime numbers and the only element in \operatorname {Irr}_{\pi '}(G) is the principal character, then G=1.
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