Quadratic characters in groups of odd order (original) (raw)

Character values of groups of odd order and a question of Feit

Journal of Algebra, 1981

Let x be an irreducible complex character of a finite group G and let Q&) denote the field obtained by adjoining the values of x to the rational field Q. Let Q, denote the field obtained by adjoining a primitive mth root of unity to Q. We say that x requires mth roots of unity if Qk) is contained in Q, and m is the smallest positive integer with this property. The following open question is raised by Feit 12, p. 411. Suppose that x requires mth roots of unity. Is it true that G contains an element of order m? The purpose of this paper is to provide an affirmative answer to the question in the case that G has odd order.

Rational irreducible characters and rational conjugacy classes in finite groups

Transactions of the American Mathematical Society, 2007

We prove that a finite group G G has two rational-valued irreducible characters if and only if it has two rational conjugacy classes, and determine the structure of any such group. Along the way we also prove a conjecture of Gow stating that any finite group of even order has a non-trivial rational-valued irreducible character of odd degree.

A Characterisation of Certain Finite Groups of Odd Order

Mathematical Proceedings of the Royal Irish Academy, 2011

The commutativity degree of a finite group is the probability that two randomly chosen group elements commute. The main object of this paper is to obtain a characterization for all finite groups of odd order with commutativity degree greater than or equal to 11 75 .

On character values in finite groups

Bulletin of the Australian Mathematical Society, 1977

Let u be a nonidentity element of a finite group G and let c be a complex number. Suppose that every nonprincipal irreducible character X of G satisfies either X(l) -X(u) = c or X(u) = 0 . It is shown that c is an even positive integer and all such groups with a -8 are described.

Nonsolvable Groups All of Whose Character Degrees are Odd-Square-Free

Communications in Algebra, 2011

A finite group G is odd-square-free if no irreducible complex character of G has degree divisible by the square of an odd prime. We determine all odd-square-free groups G satisfying S G Aut(S) for a finite simple group S. More generally, we show that if G is any nonsolvable odd-square-free group, then G has at most two nonabelian chief factors and these must be simple odd-square-free groups. If the alternating group A 7 is involved in G, the structure of G can be further restricted. was A 7. The following theorem shows that a number of other simple groups are odd-square-free. Except for A 7 , however, the simple groups S that arise all have the property that |S| is OSF. Theorem A. Let S be a finite simple group and let S H Aut(S). The group H is odd-squarefree if and only if one of the following holds: i. S ∼ = J 1. ii. S ∼ = A 5 or S ∼ = A 7. iii. S ∼ = PSL 2 (p), where p 5 is an odd prime and p 2 − 1 is odd-square-free. iv. S ∼ = PSL 2 (q), where q = 2 a , a 2 and (q 2 − 1)|H : S| is odd-square-free. v. S ∼ = 2 B 2 (q 2), where q 2 = 2 2m+1 , m 1, and (q 2 − 1)(q 4 + 1)|H : S| is square-free. The group J 1 is the first sporadic Janko group and Aut(J 1) = J 1. If S is one of A 5 , A 7 , or PSL 2 (p) with p odd, then | Aut(S) : S| = 2. Moreover, (p−1, p+1) = 2, hence p 2 −1 = (p−1)(p+1) is OSF if and only if both of p − 1 and p + 1 are OSF. If S is PSL 2 (q), q = 2 a , then the outer automorphism group of S is cyclic of order a. Thus

A characterization of certain finite groups of odd order

Mathematical Proceedings of the Royal Irish …, 2011

Extendible characters and monomial groups of odd order

Journal of Algebra, 2006

Let G be a finite p-solvable group, where p is an odd prime. We establish a connection between extendible irreducible characters of subgroups of G that lie under monomial characters of G and nilpotent subgroups of G. We also provide a way to get "good" extendible irreducible characters inside subgroups of G. As an application, we show that every normal subgroup N of a finite monomial odd p, q-group G, that has nilpotent length less than or equal to 3, is monomial. 2. Proof of Theorem B We begin with an equivalent form of Theorem 3.2 in [2]. Theorem 2.1. Suppose that F is a finite field of odd characteristic p, that G is a finite p-solvable group, that H is a subgroup of p-power index in G, that B is an anisotropic symplectic FG-module and that S is an FG-submodule of B. Then the G-invariant symplectic form on B restricts to a G-invariant symplectic form on S. If S, with this form, restricts to a hyperbolic symplectic FH-module S| H , then S = 0.

Powers of irreducible characters and conjugacy classes in finite groups

Journal of Algebra and Its Applications, 2014

In this paper, number of conjugacy classes and irreducible characters in a non-abelian group of order 2 6 are investigated using cycle pattern of elements. Through the exploits of commutator and representation of elements as a product of disjoint cycles, the number of conjugacy classes is obtained which extends some results in literature. Proposition 2.1. Any permutation of a finite set containing at least two elements can be written as the product of transpositions.

On characters in the principal 2-block, II

Journal of the Australian Mathematical Society, 1979

Let k be a non-zero complex number and let u and v be elements of a finite group G. Suppose that at most one of u and v belongs to O(G), the maximal normal subgroup of G of odd order. It is shown that G satisfies X(v) -X(u) = k for every complex nonprincipal irreducible character X in the principal 2-block of G, if and only if G/O(G) is isomorphic to one of the following groups: C t , PSL(2, 2") or PSi(2, 5 2o+1 ), where n>2 and a> 1.

Strong involutions in finite special linear groups of odd characteristic

Journal of Algebra

Let t be an involution in GL(n, q) whose fixed point space E + has dimension k between n/3 and 2n/3. For each g ∈ GL(n, q) such that tt g has even order, tt g contains a unique involution z(g) which commutes with t. We prove that, with probability at least c/ log n (for some c > 0), the restriction z(g) |E+ is an involution on E + with fixed point space of dimension between k/3 and 2k/3. This result has implications in the analysis of the complexity of recognition algorithms for finite classical groups in odd characteristic. We discuss how similar results for involutions in other finite classical groups would solve a major open problem in our understanding of the complexity of constructing involution centralisers in those groups.

Quadratic characters in groups of odd order (original) (raw)

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