David Chillag

On the length of the conjugacy classes of finite groups

Abstract Let G be a finite group. The question of how certain arithmetical conditions on the degrees of the irreducible characters of G influence the group structure was studied by several authors. Our purpose here is to impose analogous conditions on the lengths of the conjugacy classes of G and to describe the group structure under these conditions.

Character degree graphs that are complete graphs

Let G be a finite group, and write cd(G) for the set of degrees of irreducible characters of G. We define F(G) to be the graph whose vertex set is cd(G) - {1}, and there is an edge between a and b if (a, b) > 1. We prove that if Γ(G) is a complete graph, then G is a solvable group.

GENERALIZED FROBENIUS GROUPS. II

A pair (G. K) in whichG is a finite group andK◃G, 1<K<G, is said to satisfy (F2) if |CG(x)|=|CG/K(xK)| for allx∈G/K. First we survey all the examples known to us of such pairs in whichG is neither ap-group nor a Frobenius group with Frobenius kernelK. Then we show that under certain restrictions there are, essentially, all the possible examples.

Regular representations of semisimple algebras, separable field extensions, group characters, generalized circulants, and generalized cyclic codes

Let A be a semisimple, n-dimensional, commutative algebra over a field F. Fix a basis B of A, and denote by M(a; B) the transpose of the matrix over F that represents a ϵ A regularly with respect to B. It is easy to see that the set {M(a; B) |a ϵ A} can be simultaneously diagonalized over many fields (including all perfect fields). We use this fact in order to give an elementary proof that such an algebra over an infinite field is generated by a single element, and to describe the subalgebras of A in terms of certain partitions of the set {1,2,3,…, n}. Several applications of these results are shown: (1) We give a new proof for the theorem stating that every finite-dimensional, separable field extension has a primitive element. (2) We show that every finite group G has a character θ such that every other generalized character of G is a polynomial in θ with rational coefficients. (This is true for Brauer characters as well.) (3) We give a necessary condition for two generalized characters (or Brauer characters) ζ and χ that forces the field of values of ζ to contain that of χ. (4) Many collections of patterned matrices over a field F, such as circulant matrices and some of their generalizations are known to be algebras generated by a single matrix. We observe that each subalgebra of such a collection is also generated by a single matrix. Also, if a and b are two elements of such a collection, we give a necessary and sufficient condition, in terms of the eigenvalue pattern of a and b, for a to be a polynomial in b with coefficients in F. (5) We show that if A is a (generalized) cyclic code, then the eigenvalues of M(a; B) are the so-called Matteson-Solomon coefficients of the codeword a. Other applications to coding, to groups, and to field extensions are discussed as well.

Sylow-metacyclic groups andQ-admissibility

A finite groupG isQ-admissible if there exists a division algebra finite dimensional and central overQ which is a crossed product forG. AQ-admissible group is necessarily Sylow-metacyclic (all its Sylow subgroups are metacyclic). By means of an investigation into the structure of Sylow-metacyclic groups, the inverse problem (is every Sylow-metacyclic groupQ-admissible?) is essentially reduced to groups of order 2a 3b and to a list of known “almost simple” groups.

Nearly odd-order and nearly real finite groups

Two families of groups close to groups of odd order, and two families of groups close to real groups will be described. The first two are the family of finite groups in which all real irreducible ordinary characters are linear, and the family of all finite groups with the dual condition on the conjugacy classes, namely, groups in which all real conjugacy classes are contained in the center of the group. We will see that each group in one of these families is a direct product of a group of odd order with a 2-group. The families close to real groups are the family of finite groups in which every non-real irreducible ordinary character is linear, and dually, the family of all finite groups in which every non-real conjugacy class is contained in the center. For a group G in the first family we show that the collection of real elements R(G) is a normal subgroupG has a normal π-complement where π = π (∣G: R(G)∣), and a Hall π-subgroup which is either abelian or a nearly real 2-group. A group is in the second fa...

A criterion for the existence of normal π-complements in finite groups

Abstract The purpose of this article is to prove, using the classification of the finite simple groups, the following conjecture: Let π be a set of odd primes, then a finite group is π-homogeneous if and only if it is π′-closed. Using this, several open problems can be settled, including an affirmative answer to the following problem of Baer: Let G be a finite group and π ⊆ π ( G ). Suppose that G is both π-homogeneous and π′-homogeneous. Is G a direct product of a π-group and a π′-group? Finally, we note that the proof of the conjecture yields proof to some theorems proved earlier without using the classification of the finite simple groups.

Powers of Characters of Finite Groups

LetG be a finite group and θ a complex character ofG. Define Irr(θ) to be the set of all irreducible constituents of θ andIrr(G) to be the set of all irreducible characters ofG. Thecharacter-covering number of a finite groupG, ccn(G), is defined as the smallest positive integer m such thatIrr(χm) =Irr(G) for allχ∈Irr(G)—{1G}. If no such positive integer exists we say that the character-covering-number ofG is infinite. In this article we show that a finite nontrivial groupG has a finite character-covering-number if and only ifG is simple and non-abelian and ifG is a nonabelian simple group thenccn(G) ⩽ k2 − 3k + 4, wherek is the number of conjugacy classes ofG. Then we show (using the classification of the finite simple groups) that the only finite group with a character-covering-number equal to two is the smallest Jankos group,J1. These results are analogous to results obtained previously concerning the covering of groups by powers of conjugacy classes. Other related results are shown.

Semi-rational solvable groups

Abstract An element x of a finite group G is called rational if all generators of the group 〈x〉 are contained in a single conjugacy class. If all elements of G are rational, then G itself is called rational. It was proved by Gow that if G is a rational solvable group then π(G) ⊂ {2, 3, 5}. We call x ∈ G semi-rational if all generators of 〈x〉 are contained in a union of two conjugacy classes. Furthermore, we call x ∈ G inverse semi-rational if every generator of 〈x〉 is conjugate to either x or x –1. Then G is called semi-rational (resp. inverse semi-rational) if all elements of G are semi-rational (resp. inverse semi-rational). We show that if G is semi-rational and solvable then π(G) ⊂ {2, 3, 5, 7, 13, 17}, and if G is inverse semi-rational and solvable then 17 ∉ π(G). If G has odd order, then it is semi-rational if and only if it is inverse semi-rational. In this case we describe the structure of G.

Classification of Finite Groups by a Maximal Subgroup

Let A be a maximal subgroup of a finite group G. The question of how the structure of A affects the structure of G was considered in several papers. If A is nilpotent of odd order then G is solvable by a theorem of Thompson [ 10, p. 3401. If A is nilpotent and G is nonsolvable, then the groups G were classified by Baumann and Rose [4, 241. Assuming a weak version of the Unbalanced Group Theorem (WUGT) whose proof is now complete, Arad, Herzog and Shaki characterized groups in which F(G) = 1 and A contains a nilpotent subgroup H of even order with 1 A : HI < 2 (see [l] for the exact statement, which is more general). Using the Arad-Herzog-Shaki’s result and using the (WUGT) (to be stated later), we prove: