Nilpotent by Supersolvable M-Groups

1985 ◽  
Vol 37 (5) ◽  
pp. 934-962 ◽  
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.

scholarly journals Blocks and Normal Subgroups of Finite Groups

1963 ◽  
Vol 22 ◽  
pp. 15-32 ◽  
Author(s):  
W. F. Reynolds
Keyword(s):  
Finite Group ◽  
Normal Subgroup ◽  
Finite Groups ◽  
Irreducible Character ◽  
Normal Subgroups ◽  
Irreducible Characters ◽  
Section 8 ◽  
A Finite Group

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.

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.

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

1971 ◽  
Vol 23 (5) ◽  
pp. 857-865 ◽  
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 ◽  
A Finite Group

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.

Isomorphic Subgroups of Finite p-Groups. II

1971 ◽  
Vol 23 (6) ◽  
pp. 1023-1039 ◽  
Author(s):  
George Glauberman
Keyword(s):  
Finite Group ◽  
Normal Subgroup ◽  
Normal Subgroups ◽  
Nilpotence Class ◽  
A Finite Group

Suppose that we are given an isomorphism ϕ between two subgroups of index p in a finite p-group P. Let N(ϕ) be the largest subgroup of P fixed by ϕ. By a result of Sims [2, Proposition 2.1], n(ϕ) is a normal subgroup of P. In [2], we showed that P/N(ϕ) has nilpotence class at most two if p = 2, and at most three if p is odd. We then applied this result to investigate certain cases of the following question. Suppose that P is contained in a finite group G and that some subgroup of index p in P is a normal subgroup of G. Let α be an automorphism of P. Then, does α fix some nonidentity normal subgroup of P that is normal in G?In this paper, we consider characteristic subgroups of P rather than normal subgroups.

Focal Series in Finite Groups

1953 ◽  
Vol 5 ◽  
pp. 477-497 ◽  
Author(s):  
D. G. Higman
Keyword(s):  
Finite Group ◽  
Normal Subgroup ◽  
Finite Groups ◽  
Normal Subgroups ◽  
A Finite Group

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

1969 ◽  
Vol 21 ◽  
pp. 418-429 ◽  
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.

scholarly journals Structure of intersection graphs

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 ◽  
A Finite Group

<p style="text-align: left;" dir="ltr"> Let <em>G</em> be a finite group and let <em>N</em> be a fixed normal subgroup of <em>G</em>.  In this paper, a new kind of graph on <em>G</em>, 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 <em>G</em> and two distinct vertices are adjacent if their intersection in <em>N</em>. 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. </p>

COMPONENTS AND MINIMAL NORMAL SUBGROUPS OF FINITE AND PSEUDOFINITE GROUPS

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 ◽  
A Finite Group

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

2014 ◽  
Vol 57 (3) ◽  
pp. 648-657 ◽  
Author(s):  
Juping Tang ◽  
Long Miao
Keyword(s):  
Finite Group ◽  
Normal Subgroup ◽  
Maximal Subgroup ◽  
Finite Groups ◽  
Sylow Subgroup ◽  
Proper Subgroup ◽  
Normal Subgroups ◽  
Chief Factors ◽  
A Finite Group

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).

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