Average dimension of fixed point spaces with applications

作者：Guralnick Robert M^*; Maroti Attila

来源：Advances in Mathematics, 2011, 226(1): 298-308.

DOI：10.1016/j.aim.2010.06.023

• Linear representation
• Fixed point space
• Centralizer
• Chief factor
• BFC group
• Primitive character

Let G be a finite group, F a field, and V a finite dimensional FG-module such that G has no trivial composition factor on V. Then the arithmetic average dimension of the fixed point spaces of elements of G on V is at most (1/p) dim V where p is the smallest prime divisor of the order of G. This answers and generalizes a 1966 conjecture of Neumann which also appeared in a paper of Neumann and Vaughan-Lee and also as a problem in The Kourovka Notebook posted by Vaughan-Lee. Our result also generalizes a recent theorem of Isaacs, Keller, Meierfrankenfeld, and Moret6. We also classify precisely when equality can occur. Various applications are given. For example, another conjecture of Neumann and Vaughan-Lee is proven and some results of Segal and Shalev are improved and/or generalized concerning BFC groups.

• 出版日期2011-1-15