Schur indices of perfect groups

It has been noticed by many authors that the Schur indices of the irreducible characters of many quasi-simple finite groups are at most . A conjecture has emerged that the Schur indices of all irreducible characters of all quasi-simple finite groups are at most . We prove that this conjecture cannot be extended to the set of all finite perfect groups. Indeed, we prove that, given any positive integer , there exist irreducible characters of finite perfect groups of chief length which have Schur index .

     

Block separation in solvable groups

If π is a set of primes, a finite group G is block π-separated if for every two distinct irreducible complex characters α, β ∈ Irr(G) there exists a prime p ∈ π such that α and β lie in different Brauer p-blocks. A group G is block separated if it is separated by the set of prime divisors of |G|. Given a set π with n different primes, we construct an example of a solvable π-group G which is block separated but it is not separated by every proper subset of π. Received: 22 December 2004     

Character Quotients for Coprime Acting Groups

Let the finite group A be acting on a finite group G with (|A|,|G|)=1. Let be the semidirect product of A and G. Let be acharacter of irreducible after restriction to G. In a previouspaper by Brian Hartley and the author, we proved that the restrictionof to S belongs to the set C(S) obtained by running over all that arise in this manner, by assuming, in addition, that Gis a product of extraspecial groups. This was proved in general,assuming only some condition on the Green functions of groupsof Lie type that is not as yet fully verified. In the presentpaper, we define the map Q(): SC by Q()(s)=|C_G(s)|/(s). We provethat Q()C(S) under the same hypotheses. In particular, the characterquotient Q() is an ordinary character.     

The Brauer–Clifford group

Clifford theory provides well behaved character correspondences between different groups which have isomorphic quotients. Given one such quotient group, we define the Brauer–Clifford group. We show that each character of the original groups gives rise to a specific element of the Brauer–Clifford group. When two characters of different groups yield the same element of the Brauer–Clifford group, we obtain a very well behaved character correspondence between the characters of the different groups, which preserves not only induction, restriction, multiplicities, but also fields of values for the corresponding characters, and Schur indices. We also show that the Brauer–Clifford group has a natural homomorphism into a Brauer group. The Brauer–Clifford group can be thought of as a refinement of the previously introduced Clifford classes.     

Clifford classes for isoclinic groups

Let G be a finite group, N a normal subgroup of G, and an irreducible character of G. Clifford Theory studies a whole collection of related irreducible characters of all the subgroups of G that contain N. The relationships among these characters as well as their Schur indices are controlled by the Clifford class c Clif(G/N, F) of with respect to N over some field F. This is an equivalence class of central simple G/N-algebras. Assume now that G/N is cyclic. One can obtain a new isoclinic group and character by multiplying each element of each coset of N in G by an appropriate power of a fixed root of unity . We show that there is a simple formula to calculate the Clifford class of in terms of c and . Hence, the Clifford class c controls not only the Schur index of the characters of all the subgroups of G that contain N, it also controls the Schur indices of the characters of the corresponding characters of the isoclinic groups When is a |G/N|-th root of 1, our formula shows that then When = i and |G/N| = 2, the implicit transformation on Clif(Z/2Z, F) yields a group homomorphism of the group structure introduced on the Brauer-Wall group of F to describe the Schur indices of all the irreducible characters of the double covers of the symmetric and alternating groups.Received: 17 August 2001     

Characters, fields, Schur indices and divisibility

Let G be a finite nilpotent group. Suppose that G ₀ is a subgroup of G and that ${\psi}$ is an irreducible character of G ₀. Consider the set S whose elements are the natural numbers $${\rm m}_{\bf Q}(\chi)[{\bf Q}(\chi) : {\bf Q}]$$ as ${\chi}$ runs through the irreducible characters of G which contain ${\psi}$ as a summand when restricted to G ₀. Here m _Q (χ) is, as usual, the rational Schur index of ${\chi}$ , and ${[{\bf Q}(\chi) : {\bf Q}]}$ is the degree of the extension of the field of values of the character as an extension of the rationals. We prove that then the minimum element of S divides all the other elements of S. The result is not true when G is an arbitrary finite group. We also consider some variations of this result.     

Odd character correspondences in solvable groups

Let G be a finite solvable group, p be some prime, let P be a Sylow p-subgroup of G, and let N be its normalizer in G. Assume that N has odd order. Then, we prove that there exists a bijection from the set of all irreducible characters of G of degree prime to p to the set of all the irreducible characters of degree prime to p of N such that it preserves ± the degree modulo p, the field of values, and the Schur index over every field of characteristic zero. This strengthens a more general recent result [A. Turull, Character correspondences in solvable groups, J. Algebra 295 (2006) 157–178], but only for the case under consideration here. In addition, we prove some other strong character correspondences that have very good rationality properties. As one consequence, we prove that a solvable group G has a non-trivial rational irreducible character with degree prime to p if and only if the order of the normalizer of a Sylow p-subgroup of G has even order.