Blocks and Normal Subgroups of Finite Groups

Let H be a normal subgroup of a finite group G, and let ζ be an (absolutely) irreducible character of H. In [7], Clifford studied the irreducible characters X of G whose restrictions to H contain ζ as a constituent. First he reduced this question to the same question in the so-called inertial subgroup S of ζ in G, and secondly he described the situation in S in terms of certain projective characters of S/H. In section 8 of [10], Mackey generalized these results to the situation where all the characters concerned are projective.

Nilpotent by Supersolvable M-Groups

10.4153/cjm-1985-051-0 ◽

1985 ◽

Vol 37 (5) ◽

pp. 934-962 ◽

Cited By ~ 8

Author(s):

Alan E. Parks

Keyword(s):

Finite Group ◽

Normal Subgroup ◽

Normal Subgroups ◽

Restrictive Assumption ◽

Irreducible Characters ◽

Even Order ◽

A Finite Group ◽

Prime 2

A character of a finite group G is monomial if it is induced from a linear (degree one) character of a subgroup of G. A group G is an M-group if all its complex irreducible characters (the set Irr(G)) are monomial.In [1], Dade gave an example of an M-group with a normal subgroup which is itself not an M-group. In his group G, the supersolvable residual N is an extra special 2-group and G/N is supersolvable of even order. Moreover, the prime 2 is used in such a way that no analogous construction is possible in the case that |N| or |G:N| is odd. This led Isaacs in [8] and Dade in [2] to consider the effect of certain “oddness“ hypotheses in the study of monomial characters.Our main results are in the same spirit. Although our techniques seem to require a restrictive assumption on the supersolvable residual of the groups we consider, our theorems provide more evidence that under fairly general circumstances normal subgroups of M-groups should be M-groups.

Irreducible characters of even degree and normal Sylow 2-subgroups

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004116000669 ◽

2016 ◽

Vol 162 (2) ◽

pp. 353-365 ◽

Cited By ~ 1

Author(s):

NGUYEN NGOC HUNG ◽

PHAM HUU TIEP

Keyword(s):

Finite Group ◽

Finite Groups ◽

Irreducible Character ◽

Average Degree ◽

Character Degrees ◽

Irreducible Characters ◽

AbstractThe classical Itô-Michler theorem on character degrees of finite groups asserts that if the degree of every complex irreducible character of a finite group G is coprime to a given prime p, then G has a normal Sylow p-subgroup. We propose a new direction to generalize this theorem by introducing an invariant concerning character degrees. We show that if the average degree of linear and even-degree irreducible characters of G is less than 4/3 then G has a normal Sylow 2-subgroup, as well as corresponding analogues for real-valued characters and strongly real characters. These results improve on several earlier results concerning the Itô-Michler theorem.

A Dual View of the Clifford Theory of Characters of Finite Groups

10.4153/cjm-1971-096-6 ◽

1971 ◽

Vol 23 (5) ◽

pp. 857-865 ◽

Cited By ~ 8

Author(s):

Richard L. Roth

Keyword(s):

Finite Group ◽

Normal Subgroup ◽

Finite Groups ◽

Essential Role ◽

Dual Group ◽

Complex Character ◽

Irreducible Characters ◽

Intermediate Subgroup ◽

Clifford Theory ◽

Let G be a finite group, K a normal subgroup of G, χ an irreducible complex character of G. In the usual decomposition of χ|κ, using Clifford's theorems, G/K is seen to operate by conjugation on the irreducible characters of K and if σ is an irreducible component of χ|κ, then I(σ) the inertial group of σ, plays an essential role as an appropriate intermediate subgroup for the analysis. In this paper we consider the case where G/K is abelian and study the action of the dual group (G/K)^ (of linear characters of G/K) on the irreducible characters of G effected by multiplication. This action appears to be related in a dual way to the action of G/K on the characters of K. We define a subgroup J(χ) of G which plays a role similar to that of I (σ) and which we call the dual inertial group of χ.

Focal Series in Finite Groups

10.4153/cjm-1953-055-5 ◽

1953 ◽

Vol 5 ◽

pp. 477-497 ◽

Cited By ~ 17

Author(s):

D. G. Higman

Keyword(s):

Finite Group ◽

Normal Subgroup ◽

Finite Groups ◽

Normal Subgroups ◽

If there is given a subgroup 5 of a (finite) group G, we may ask what information is to be obtained about the structure of G from a knowledge of the location of S in G. Thus, for example, famed theorems of Frobenius and Burnside give criteria for the existence of a normal subgroup N of G such that G = NS and 1 = N ⋂ S, and hence in particular for the non-simplicity of G. To aid in locating S in G, and to facilitate exploitation of the transfer, we single out a descending chain of normal subgroups of S. Namely, we introduce the focal series of S in G by means of the recursive formulae

Generalized Frattini Subgroups of Finite Groups. II

10.4153/cjm-1969-046-3 ◽

1969 ◽

Vol 21 ◽

pp. 418-429 ◽

Cited By ~ 2

Author(s):

James C. Beidleman

Keyword(s):

Finite Group ◽

Normal Subgroup ◽

Finite Groups ◽

Proper Subgroup ◽

Subnormal Subgroup ◽

Normal Subgroups ◽

Frattini Subgroup ◽

A Finite Group ◽

Proper Normal Subgroup ◽

Equivalent Conditions

The theory of generalized Frattini subgroups of a finite group is continued in this paper. Several equivalent conditions are given for a proper normal subgroup H of a finite group G to be a generalized Frattini subgroup of G. One such condition on H is that K is nilpotent for each normal subgroup K of G such that K/H is nilpotent. From this result, it follows that the weakly hyper-central normal subgroups of a finite non-nilpotent group G are generalized Frattini subgroups of G.Let H be a generalized Frattini subgroup of G and let K be a subnormal subgroup of G which properly contains H. Then H is a generalized Frattini subgroup of K.Let ϕ(G) be the Frattini subgroup of G. Suppose that G/ϕ(G) is nonnilpotent, but every proper subgroup of G/ϕ(G) is nilpotent. Then ϕ(G) is the unique maximal generalized Frattini subgroup of G.

Structure of intersection graphs

Indonesian Journal of Combinatorics ◽

10.19184/ijc.2021.5.2.6 ◽

2021 ◽

Vol 5 (2) ◽

pp. 102

Author(s):

Haval M. Mohammed Salih ◽

Sanaa M. S. Omer

Keyword(s):

Finite Group ◽

Normal Subgroup ◽

Finite Groups ◽

Special Linear Group ◽

Intersection Graph ◽

Graph Structure ◽

Normal Subgroups ◽

Intersection Graphs ◽

Cyclic Groups ◽

Let G be a finite group and let N be a fixed normal subgroup of G. In this paper, a new kind of graph on G, namely the intersection graph is defined and studied. We use <img src="/public/site/images/ikhsan/equation.png" alt="" width="6" height="4" /> to denote this graph, with its vertices are all normal subgroups of G and two distinct vertices are adjacent if their intersection in N. We show some properties of this graph. For instance, the intersection graph is a simple connected with diameter at most two. Furthermore we give the graph structure of <img src="/public/site/images/ikhsan/equation_(1).png" alt="" width="6" height="4" /> for some finite groups such as the symmetric, dihedral, special linear group, quaternion and cyclic groups.

COMPONENTS AND MINIMAL NORMAL SUBGROUPS OF FINITE AND PSEUDOFINITE GROUPS

Journal of Symbolic Logic ◽

10.1017/jsl.2018.71 ◽

2019 ◽

Vol 84 (1) ◽

pp. 290-300

Author(s):

JOHN S. WILSON

Keyword(s):

Finite Group ◽

Group Theory ◽

Normal Subgroup ◽

Finite Groups ◽

Minimal Normal Subgroup ◽

Normal Subgroups ◽

First Order ◽

Order Language ◽

Image Position ◽

AbstractIt is proved that there is a formula$\pi \left( {h,x} \right)$in the first-order language of group theory such that each component and each non-abelian minimal normal subgroup of a finite groupGis definable by$\pi \left( {h,x} \right)$for a suitable elementhofG; in other words, each such subgroup has the form$\left\{ {x|x\pi \left( {h,x} \right)} \right\}$for someh. A number of consequences for infinite models of the theory of finite groups are described.

On the ℱϕ-Hypercentre of Finite Groups

Canadian Mathematical Bulletin ◽

10.4153/cmb-2014-021-9 ◽

2014 ◽

Vol 57 (3) ◽

pp. 648-657 ◽

Cited By ~ 3

Author(s):

Juping Tang ◽

Long Miao

Keyword(s):

Finite Group ◽

Normal Subgroup ◽

Maximal Subgroup ◽

Finite Groups ◽

Sylow Subgroup ◽

Proper Subgroup ◽

Normal Subgroups ◽

Chief Factors ◽

AbstractLet G be a finite group and let ℱ be a class of groups. Then Zℱϕ(G) is the ℱϕ-hypercentre of G, which is the product of all normal subgroups of G whose non-Frattini G-chief factors are ℱ-central in G. A subgroup H is called ℳ-supplemented in a finite group G if there exists a subgroup B of G such that G = HB and H1B is a proper subgroup of G for any maximal subgroup H1 of H. The main purpose of this paper is to prove the following: Let E be a normal subgroup of a group G. Suppose that every noncyclic Sylow subgroup P of F*(E) has a subgroup D such that 1 < |D| < |P| and every subgroup H of P with order |H| = |D| is 𝓜-supplemented in G, then E ≤ Zuϕ(G).

On c#-normal subgroups of finite groups

Journal of Algebra and Its Applications ◽

10.1142/s0219498817501602 ◽

2016 ◽

Vol 16 (08) ◽

pp. 1750160

Author(s):

Guo Zhong ◽

Shi-Xun Lin

Keyword(s):

Finite Group ◽

Normal Subgroup ◽

Finite Groups ◽

Sufficient Conditions ◽

Necessary And Sufficient Conditions ◽

Normal Subgroups ◽

A Finite Group ◽

Necessary And Sufficient

Let [Formula: see text] be a subgroup of a finite group [Formula: see text]. We say that [Formula: see text] is a [Formula: see text]-normal subgroup of [Formula: see text] if there exists a normal subgroup [Formula: see text] of [Formula: see text] such that [Formula: see text] and [Formula: see text] is a [Formula: see text]-subgroup of [Formula: see text]. In the present paper, we use [Formula: see text]-normality of subgroups to characterize the structure of finite groups, and establish some necessary and sufficient conditions for a finite group to be [Formula: see text]-supersolvable, [Formula: see text]-nilpotent and solvable. Our results extend and improve some recent ones.

PARTIAL COMPLEMENTS IN FINITE GROUPS

Journal of the Australian Mathematical Society ◽

10.1017/s1446788710000248 ◽

2010 ◽

Vol 89 (1) ◽

pp. 1-7

Author(s):

INGRID CHEN

Keyword(s):

Finite Group ◽

Normal Subgroup ◽

Finite Groups ◽

Normal Subgroups ◽

AbstractLet G be a finite group with normal subgroup N. A subgroup K of G is a partial complement of N in G if N and K intersect trivially. We study the partial complements of N in the following case: G is soluble, N is a product of minimal normal subgroups of G, N has a complement in G, and all such complements are G-conjugate.