scholarly journals On the determination of the ramification index in Clifford's theorem

1988 ◽  
Vol 31 (3) ◽  
pp. 469-474
Author(s):  
Robert W. van der Waall
Keyword(s):  
Finite Group ◽  
Normal Subgroup ◽  
Ramification Index ◽  
A Finite Group

Let K be a field, G a finite group, V a (right) KG-module. If H is a subgroup of G, then, restricting the action of G on V to H, V is also a KH-module. Notation: VH.Suppose N is a normal subgroup of G. The KN-module VN is not irreducible in general, even when V is irreducible as KG-module. A part of the well-known theorem of A. H. Clifford [1, V.17.3] yields the following.

ON THE REGULAR GRAPH RELATED TO THE G-CONJUGACY CLASSES

2021 ◽  
pp. 1-5
Author(s):  
SH. RAHIMI ◽  
Z. AKHLAGHI
Keyword(s):  
Finite Group ◽  
Normal Subgroup ◽  
Conjugacy Class ◽  
Regular Graph ◽  
Simple Graph ◽  
Conjugacy Classes ◽  
A Finite Group

Abstract Given a finite group G with a normal subgroup N, the simple graph $\Gamma _{\textit {G}}( \textit {N} )$ is a graph whose vertices are of the form $|x^G|$ , where $x\in {N\setminus {Z(G)}}$ and $x^G$ is the G-conjugacy class of N containing the element x. Two vertices $|x^G|$ and $|y^G|$ are adjacent if they are not coprime. We prove that, if $\Gamma _G(N)$ is a connected incomplete regular graph, then $N= P \times {A}$ where P is a p-group, for some prime p, $A\leq {Z(G)}$ and $\textbf {Z}(N)\not = N\cap \textbf {Z}(G)$ .

scholarly journals A on simplicity criterion for finite groups

1969 ◽  
Vol 10 (3-4) ◽  
pp. 359-362
Author(s):  
Nita Bryce
Keyword(s):  
Finite Group ◽  
Normal Subgroup ◽  
Finite Groups ◽  
Odd Order ◽  
A Finite Group

M. Suzuki [3] has proved the following theorem. Let G be a finite group which has an involution t such that C = CG(t) ≅ SL(2, q) and q odd. Then G has an abelian odd order normal subgroup A such that G = CA and C ∩ A = 〈1〉.

scholarly journals A remark on the C-normality of maximal subgroups of finite groups

1997 ◽  
Vol 40 (2) ◽  
pp. 243-246
Author(s):  
Yanming Wang
Keyword(s):  
Finite Group ◽  
Normal Subgroup ◽  
Finite Groups ◽  
Maximal Subgroups ◽  
Primitive Groups ◽  
A Finite Group

A subgroup H is called c-normal in a group G if there exists a normal subgroup N of G such that HN = G and H∩N ≤ HG, where HG =: Core(H) = ∩g∈GHg is the maximal normal subgroup of G which is contained in H. We use a result on primitive groups and the c-normality of maximal subgroups of a finite group G to obtain results about the influence of the set of maximal subgroups on the structure of G.

FINITE GROUPS WITH GIVEN NEARLY S-QUASINORMAL SUBGROUPS

2008 ◽  
Vol 01 (03) ◽  
pp. 369-382
Author(s):  
Nataliya V. Hutsko ◽  
Vladimir O. Lukyanenko ◽  
Alexander N. Skiba
Keyword(s):  
Finite Group ◽  
Normal Subgroup ◽  
Finite Groups ◽  
Sylow Subgroup ◽  
Cyclic Subgroup ◽  
Saturated Formation ◽  
Supersoluble Groups ◽  
Sylow Subgroups ◽  
Quasinormal Subgroup ◽  
A Finite Group

Let G be a finite group and H a subgroup of G. Then H is said to be S-quasinormal in G if HP = PH for all Sylow subgroups P of G. Let HsG be the subgroup of H generated by all those subgroups of H which are S-quasinormal in G. Then we say that H is nearly S-quasinormal in G if G has an S-quasinormal subgroup T such that HT = G and T ∩ H ≤ HsG. Our main result here is the following theorem. Let [Formula: see text] be a saturated formation containing all supersoluble groups and G a group with a normal subgroup E such that [Formula: see text]. Suppose that every non-cyclic Sylow subgroup P of E has a subgroup D such that 1 < |D| < |P| and all subgroups H of P with order |H| = |D| and every cyclic subgroup of P with order 4 (if |D| = 2 and P is a non-abelian 2-group) having no supersoluble supplement in G are nearly S-quasinormal in G. Then [Formula: see text].

scholarly journals Block Extensions, Local Categories and Basic Morita Equivalences

2020 ◽  
Vol 71 (2) ◽  
pp. 703-728
Author(s):  
Tiberiu Coconeţ ◽  
Andrei Marcus ◽  
Constantin-Cosmin Todea
Keyword(s):  
Finite Group ◽  
Normal Subgroup ◽  
Cohomology Class ◽  
Group Extension ◽  
Modular System ◽  
Pointed Group ◽  
A Finite Group ◽  
Nilpotent Blocks ◽  
The Way

Abstract Let $(\mathcal{K},\mathcal{O},k)$ be a $p$-modular system where $p$ is a prime and $k$ algebraically closed, let $b$ be a $G$-invariant block of the normal subgroup $H$ of a finite group $G$, having defect pointed group $Q_\delta$ in $H$ and $P_\gamma$ in $G$ and consider the block extension $b\mathcal{O}G$. One may attach to $b$ an extended local category $\mathcal{E}_{(b,H,G)}$, a group extension $L$ of $Z(Q)$ by $N_G(Q_\delta )/C_H(Q)$ having $P$ as a Sylow $p$-subgroup, and a cohomology class $[\alpha ]\in H^2(N_G(Q_\delta )/QC_H(Q),k^\times )$. We prove that these objects are invariant under the $G/H$-graded basic Morita equivalences. Along the way, we give alternative proofs of the results of Külshammer and Puig (1990), and Puig and Zhou (2012) on extensions of nilpotent blocks. We also deduce by our methods a result of Zhou (2016) on $p^{\prime}$-extensions of inertial blocks.

scholarly journals The p-Huppert-subgroup and the set of p-quasi-superfluous elements in a finite group

1993 ◽  
Vol 36 (2) ◽  
pp. 289-297
Author(s):  
Angel Carocca ◽  
Rudolf Maier
Keyword(s):  
Finite Group ◽  
Normal Subgroup ◽  
Prime Number ◽  
Special Interest ◽  
Fitting Subgroup ◽  
Supersoluble Groups ◽  
A Finite Group ◽  
Parametrized Family

Based on the theory of p-supersoluble and supersoluble groups, a prime-number parametrized family of canonical characteristic subgroups Γp(G) and their intersection Γ(G) is introduced in every finite group G and some of its properties are studied. Special interest is dedicated to an elementwise description of the largest p-nilpotent normal subgroup of Γp(G) and of the Fitting subgroup of Γ(G).

THE NUMBER OF CONJUGACY CLASSES CONTAINED IN NORMAL SUBGROUPS OF SOLVABLE GROUPS

2013 ◽  
Vol 12 (05) ◽  
pp. 1250204
Author(s):  
AMIN SAEIDI ◽  
SEIRAN ZANDI
Keyword(s):  
Finite Group ◽  
Normal Subgroup ◽  
Solvable Groups ◽  
Conjugacy Classes ◽  
Normal Subgroups ◽  
A Finite Group

Let G be a finite group and let N be a normal subgroup of G. Assume that N is the union of ξ(N) distinct conjugacy classes of G. In this paper, we classify solvable groups G in which the set [Formula: see text] has at most three elements. We also compute the set [Formula: see text] in most cases.

scholarly journals On extension of characters from normal subgroups

1984 ◽  
Vol 27 (1) ◽  
pp. 7-9 ◽  
Author(s):  
G. Karpilovsky
Keyword(s):  
Finite Group ◽  
Normal Subgroup ◽  
Sufficient Condition ◽  
Normal Subgroups ◽  
Complex Character ◽  
A Finite Group

In what follows, character means irreducible complex character.Let G be a finite group and let % be a character of a normal subgroup N. If χ extends to a character of G then χ is stabilised by G, but the converse is false. The aim of this paper is to prove the following theorem which gives a sufficient condition for χ to be extended to a character of G.

scholarly journals On characters in the principal 2-block

1978 ◽  
Vol 25 (3) ◽  
pp. 264-268 ◽  
Author(s):  
Thomas R. Berger ◽  
Marcel Herzog
Keyword(s):  
Finite Group ◽  
Normal Subgroup ◽  
Cyclic Group ◽  
Complex Number ◽  
Irreducible Character ◽  
Odd Order ◽  
A Finite Group

AbstractLet k be a complex number and let u be an element of a finite group G. Suppose that u does not belong to O(G), the maximal normal subgroup of G of odd order. It is shown that G satisfies X(1) – X(u) = k for every complex nonprincipal irreducible character X in the principal 2-block of G if and only if G/O(G) is isomorphic either to C2, a cyclic group of order 2, or to PSL (2, 2n), n ≧ 2.

Finite groups with weakly ℋC-embedded subgroups

2019 ◽  
Vol 19 (05) ◽  
pp. 2050093 ◽  
Author(s):  
M. Ramadan
Keyword(s):  
Finite Group ◽  
Normal Subgroup ◽  
Finite Groups ◽  
Normal Closure ◽  
A Finite Group

Let [Formula: see text] be a finite group and [Formula: see text] a subgroup of [Formula: see text]. We say that [Formula: see text] is an [Formula: see text]-subgroup of [Formula: see text] if [Formula: see text] for all [Formula: see text]. We say that [Formula: see text] is weakly [Formula: see text]-embedded in [Formula: see text] if [Formula: see text] has a normal subgroup [Formula: see text] such that [Formula: see text] and [Formula: see text] for all [Formula: see text] where [Formula: see text] is the normal closure of [Formula: see text] in [Formula: see text]. For each prime [Formula: see text] dividing the order of [Formula: see text] let [Formula: see text] be a Sylow [Formula: see text]-subgroup of [Formula: see text]. We fix a subgroup of [Formula: see text] of order [Formula: see text] with [Formula: see text] and study the structure of [Formula: see text] under the assumption that every subgroup of [Formula: see text] of order [Formula: see text] [Formula: see text] is weakly [Formula: see text]-embedded in [Formula: see text]. Our results improve and generalize several recent results in the literature.

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