1. (i) Suppose K is a conjugacy class of Sn contained in An; then K is called split if K is a union of two conjugacy classes of An. Show that the number of split conjugacy classes contained in An is equal to the number of characters χ ∈ Irr(Sn) such that χAn is not irreducible. (Hint. Consider the vector space of class functions on An which are invariant under conjugation by the transposition (12).)

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Character Theory of Finite Groups

Generalised norms in finite soluble groups

In this survey paper the authors specify all the known findings related to the norms of the group and their generalizations. Special attention is paid to the analysis of their own study of different generalized norms, particularly the norm of non-cyclic subgroups, the norm of Abelian non-cyclic subgroups, the norm of infinite subgroups, the norm of infinite Abelian subgroups and the norm of other systems of Abelian subgroups.

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Generalized norms of groups

Baer and Wielandt in 1934 and 1958, respectively, considered that the intersection of the normalizers of all subgroups of G and the intersection of the normalizers of all subnormal subgroups of G. In this article, for a finite group G, we define the subgroup S(G) to be intersection of the normalizers of all non-cyclic subgroups of G. Groups whose noncyclic subgroups are normal are studied in this article, as well as groups in which all noncyclic subgroups are normalized by all minimal subgroups. In particular, we extend the results of Passman, Bozikov, and Janko to non-nilpotent finite groups.

On a Generalization of Hamiltonian Groups and a Dualization of PN-Groups

Abstract The aim of this note is to classify the finite groups whose irreducible characters vanish on at most three conjugacy classes in the character table.

Finite groups whose irreducible characters vanish on at most three conjugacy classes

G, we dene the subgroup A(G) to be intersection of the normalizers of all non-cyclic subgroups of G. Set A0 = 1. Dene Ai+1(G)=Ai(G) = A(G=Ai(G)) for i 1. By A1 (G) denote the terminal term of the ascending series. It is proved that if G.

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Finite non-nilpotent generalizations of hamiltonian groups

Abstract For a finite group G and a non-linear irreducible complex character χ of G write υ(χ) = {g ∈ G|χ(g) = 0}. There are two dual questions as follows: what are the finite groups G such that υ(χ) is a conjugacy class for all but one of the non-linear irreducible characters χ of G, and what are the finite groups G whose character table has the property that all but one of the columns have at most one zero entry. We completely answer the two questions using the classification of finite simple groups.

Jinshan Zhang

Papers

Finite non-nilpotent generalizations of hamiltonian groups

On a Generalization of Hamiltonian Groups and a Dualization of PN-Groups

Two dual questions on zeros of characters of finite groups