Search results for "Finite group"
showing 10 items of 205 documents
On two classes of finite supersoluble groups
2017
ABSTRACTLet ℨ be a complete set of Sylow subgroups of a finite group G, that is, a set composed of a Sylow p-subgroup of G for each p dividing the order of G. A subgroup H of G is called ℨ-S-semipermutable if H permutes with every Sylow p-subgroup of G in ℨ for all p∉π(H); H is said to be ℨ-S-seminormal if it is normalized by every Sylow p-subgroup of G in ℨ for all p∉π(H). The main aim of this paper is to characterize the ℨ-MS-groups, or groups G in which the maximal subgroups of every Sylow subgroup in ℨ are ℨ-S-semipermutable in G and the ℨ-MSN-groups, or groups in which the maximal subgroups of every Sylow subgroup in ℨ are ℨ-S-seminormal in G.
Defect zero characters predicted by local structure
2017
Let $G$ be a finite group and let $p$ be a prime. Assume that there exists a prime $q$ dividing $|G|$ which does not divide the order of any $p$-local subgroup of $G$. If $G$ is $p$-solvable or $q$ divides $p-1$, then $G$ has a $p$-block of defect zero. The case $q=2$ is a well-known result by Brauer and Fowler.
On finite groups with many supersoluble subgroups
2017
[EN] The solubility of a finite group with less than 6 non-supersoluble subgroups is confirmed in the paper. Moreover we prove that a finite insoluble group has exactly 6 non-supersoluble subgroups if and only if it is isomorphic to A5 or SL2 (5). Furthermore, it is shown that a finite insoluble group has exactly 22 non-nilpotent subgroups if and only if it is isomorphic to A5 or SL2 (5). This confirms a conjecture of Zarrin (Arch Math (Basel) 99:201 206, 2012).
The average element order and the number of conjugacy classes of finite groups
2021
Abstract Let o ( G ) be the average order of the elements of G, where G is a finite group. We show that there is no polynomial lower bound for o ( G ) in terms of o ( N ) , where N ⊴ G , even when G is a prime-power order group and N is abelian. This gives a negative answer to a question of A. Jaikin-Zapirain.
Maximal subgroups and PST-groups
2013
A subgroup H of a group G is said r to permute with a subgroup K of G if HK is a subgroup of G. H is said to be permutable (resp. S-permutable) if it permutes with all the subgroups (resp. Sylow subgroups) of G. Finite groups in which permutability (resp. S-permutability) is a transitive relation are called PT-groups (resp. PST-groups). PT-, PST- and T-groups, or groups in which normality is transitive, have been extensively studied and characterised. Kaplan [Kaplan G., On T-groups, supersolvable groups, and maxmial subgroups, Arch. Math. (Basel), 2011, 96(1), 19-25)] presented some new characterisations of soluble T-groups. The main goal of this paper is to establish PT- and PST-versions o…
Regular orbits of actions of finite soluble groups. Applications
2019
A lo largo de esta tesis, todos los conjuntos, grupos, cuerpos y módulos considerados se suponen finitos. Consideremos un grupo G actuando sobre un conjunto no vacío Ω. Decimos que la órbita de un w ∈ Ω es regular si C G (w) = {g ∈ G : wg = w} = 1; en este caso, dicha órbita consta de |G| elementos. El estudio de órbitas regulares de grupos lineales, es decir, órbitas regulares de acciones de subgrupos de GL(V ), siendo V un espacio vectorial, es importante en el desarrollo de muchas ramas de la teoría de grupos, incluendo los grupos resolubles, teoría de representaciones y grupos de permutaciones. De hecho, la solución de algunos problemas importantes en el área como el problema k(GV ) ([2…
Some results on locally finite groups
2017
En esta tesis se presentan algunos resultados sobre p-nilpotencia y permutabilidad en grupos localmente finitos. Está estructurada en cinco capítulos. El primer capítulo, que tiene carácter introductorio: contiene definiciones y resultados conocidos que serán utilizados en los capítulos sucesivos. Por tratarse de resultados ya conocidos, se introducen con referencias y sin demostraciones. En el capítulo 2 se trata la p-nilpotencia en grupos hiperfinitos, donde p es un primo. Los resultados presentados se encuentran publicados en el siguiente artículo: Ballester-Bolinches, A.; Camp-Mora, S.; Spagnuolo, F., "On p-nilpotency of hyperfinite groups". Monatshefte f¨ur Mathematik, 176, no. 4, 497–…
Pronormal subgroups of a direct product of groups
2009
[EN] We give criteria to characterize abnormal, pronormal and locally pronormal subgroups of a direct product of two finite groups A×B, under hypotheses of solvability for at least one of the factors, either A or B.
Fixed point spaces, primitive character degrees and conjugacy class sizes
2006
Let G be a finite group that acts on a nonzero finite dimensional vector space V over an arbitrary field. Assume that V is completely reducible as a G-module, and that G fixes no nonzero vector of V. We show that some element g ∈ G has a small fixed-point space in V. Specifically, we prove that we can choose g so that dim C V (g) < (1/p)dim V, where p is the smallest prime divisor of |G|.
On the abnormal structure of finite groups
2014
We study finite groups in which every maximal subgroup is supersoluble or normal. Our results answer some questions arising from papers of Asaad and Rose.