Landau's theorem, fields of values for characters, and solvable groups

2015/06/26 by Lewis, Mark L.
#20C15 #20E45 #FOS: Mathematics #Group Theory (math.GR) #Secondary 20D10

paper · doi:10.48550/arxiv.1506.08169

Abstract

When G is solvable group, we prove that the number of conjugacy classes of elements of prime power order is less than or equal to the number of irreducible characters with values in fields where \mathbb Q is extended by prime power roots of unity. We then combine this result with a theorem of Héthelyi and Külshammer that bounds the order of a finite group in terms of the number of conjugacy classes of elements of prime power order to bound the order of a solvable group by the number of irreducible characters with values in fields extended by prime power roots of unity. This yields for solvable groups a generalization of Landau's theorem.

Related

• Landau's Theorem for pi-blocks of pi-separable groups
• A lower bound for the number of conjugacy classes of finite groups
• More on Landau's theorem and Conjugacy Classes
• A new lower bound for the number of conjugacy classes
• Multiplicities of conjugacy class sizes of finite groups