David Craven,

Mathematical Institute, 24--29 St Giles', University of Oxford, Oxford, OX1 3LB, United Kingdom.

Symmetric Group Character Degrees and Hook Numbers,

The multiplicity of a character degree n for a finite group G is the number of distinct irreducible characters whose degree is equal to n. Let m(G) denote the largest such multiplicity as n runs over all natural numbers. This paper proves that, when G is a symmetric group S_r, the number r is bounded by a function of m(S_r); that is, as r tends to infinity, m(S_r) tends to infinity. This proves the corresponding result for all finite groups, by a result of Moreto, (reference in paper).

Submitted to the LMS.