A finite group $G$ is called uniformly semi-rational if there exists an integer $r$ such that the generators of every cyclic sugroup $\langle x \rangle$ of $G$ lie in at …
A finite group $G$ is called uniformly semi-rational if there exists an integer $r$ such that the generators of every cyclic sugroup $\langle x \rangle$ of $G$ lie in at most two conjugacy classes, namely $x^G$ or $(x^r)^G$. In this paper, we provide a classification of uniformly semi-rational non-abelian simple groups with particular focus on alternating groups.
We introduce and study some families of groups whose irreducible characters take values on quadratic extensions of the rationals. We focus mostly on a generalization of inverse semi-rational groups, which …
We introduce and study some families of groups whose irreducible characters take values on quadratic extensions of the rationals. We focus mostly on a generalization of inverse semi-rational groups, which we call uniformly semi-rational groups. Moreover, we associate to every finite group two invariants, called rationality and semi-rationality of the group. They measure respectively how far a group is from being rational and how much uniformly rational it is. We determine the possible values that these invariants may take for finite nilpotent groups. We also classify the fields that can occur as the field generated by the character values of a finite nilpotent group.
Abstract A finite group whose irreducible complex characters are rational-valued is called a rational group. The aim of this paper is to determine the rational almost simple and rational quasi-simple …
Abstract A finite group whose irreducible complex characters are rational-valued is called a rational group. The aim of this paper is to determine the rational almost simple and rational quasi-simple groups.
To my mind the theorem classifying the finite simple groups is the most important result in finite group theory. As I indicated in the preface, the Classification Theorem is the …
To my mind the theorem classifying the finite simple groups is the most important result in finite group theory. As I indicated in the preface, the Classification Theorem is the foundation for a powerful theory of finite groups which proceeds by reducing suitable group theoretical questions to questions about representations of simple groups. The final chapter of this book is devoted primarily to a discussion of the Classification Theorem and the finite simple groups themselves.
By G. Higman and M. B. Powell: pp. xii, 327. £6. (Academic Press Inc.,(London)Ltd., London, 1971.)
By G. Higman and M. B. Powell: pp. xii, 327. £6. (Academic Press Inc.,(London)Ltd., London, 1971.)
(1977). Finite Simple Groups. The American Mathematical Monthly: Vol. 84, No. 9, pp. 693-714.
(1977). Finite Simple Groups. The American Mathematical Monthly: Vol. 84, No. 9, pp. 693-714.
In this lecture I shall report on the recent results, and open questions, related to the congruence subgroup problem, computation of the covolume of S-arithmetic subgroups, bounds for the class-number …
In this lecture I shall report on the recent results, and open questions, related to the congruence subgroup problem, computation of the covolume of S-arithmetic subgroups, bounds for the class-number of simply connected semi-simple groups and state the finiteness theorems of [3]. We shall also briefly mention the recent work on super-rigidity of cocompact discrete subgroups of Sp(w, 1) and the R-rank 1 form of type F4, which implies arithmeticity of these discrete subgroups.
A finite group is called semi-rational if the distribution induced on it by any word map is a virtual character. Amit and Vishne give a sufficient condition for a group …
A finite group is called semi-rational if the distribution induced on it by any word map is a virtual character. Amit and Vishne give a sufficient condition for a group to be semi-rational, and ask whether it is also necessary. We answer this in the negative, by exhibiting two new criteria for semi-rationality, each giving rise to an infinite family of semi-rational groups which do not satisfy the Amit-Vishne condition. On the other hand, we use recent work of Lubotzky to show that for finite simple groups the Amit-Vishne condition is indeed necessary, and we use this to construct the first known example of an infinite family of non-semi-rational groups.
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 …
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.
The aim of this article is to explore global and local properties of finite groups whose integral group rings have only trivial central units, so-called cut groups. For such a …
The aim of this article is to explore global and local properties of finite groups whose integral group rings have only trivial central units, so-called cut groups. For such a group we study actions of Galois groups on its character table and show that the natural actions on the rows and columns are essentially the same, in particular the number of rational-valued irreducible characters coincides with the number of rational-valued conjugacy classes. Further, we prove a natural criterion for nilpotent groups of class 2 to be cut and give a complete list of simple cut groups. Also, the impact of the cut property on Sylow 3-subgroups is discussed. We also collect substantial data on groups which indicates that the class of cut groups is surprisingly large. Several open problems are included.