# Degrees, class sizes and divisors of character values

@article{Gallagher2012DegreesCS,
title={Degrees, class sizes and divisors of character values},
author={Patrick X. Gallagher},
journal={Journal of Group Theory},
year={2012},
volume={15},
pages={455 - 467}
}
• P. Gallagher
• Published 1 July 2012
• Mathematics
• Journal of Group Theory
Abstract. In the character table of a finite group there is a tendency either for the character degree to divide the conjugacy class size or the character value to vanish. There is also a partial divisibility where the determinant of the character is not 1. There are versions of these depending on a subgroup, based on an arithmetic property of spherical functions which generalizes the integrality of the values of the characters and the central characters.
6 Citations
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## References

SHOWING 1-10 OF 24 REFERENCES

• Mathematics
• 2010
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
• Mathematics
• 2004
Letm be a fixed non-negative integer. In this work we try to answer the following question: What can be said about a (finite) groupG if all of its irreducible (complex) characters vanish on at mostm
Let \chi be an irreducible character of the finite group G. If g is an element of G and \chi(g) is not zero, then we conjecture that the order of g divides |G|/\chi(1). The conjecture is a
• Mathematics
• 1999
In this paper, we consider elements x of a finite group G with the property that χ(x) ≠ 0 for all irreducible characters χ of G. If G is solvable and x has odd order, we show that x must lie in the
We give a new formula for the values of an irreducible character of the symmetric group S_n indexed by a partition of rectangular shape. Some observations and a conjecture are given concerning a
• I. Isaacs
• Mathematics