Vol. 23, No. 2, 1967

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Finite groups with small character degrees and large prime divisors

I. Martin (Irving) Isaacs

Vol. 23 (1967), No. 2, 273–280

In earlier papers of this author and D. S. Passman, some properties of finite groups with r.b.n were discussed, where we say that a group G has r.b. n (representation bound n) if all the absolutely irreducible characters of G have degree n. In the present paper, the situation where p||G| is a prime which in some sense is large when compared with n is explored. An earlier result of this nature to which we refer states that if G has r.b. (p 1) then an Sp subgroup of G is normal and abelian. Here we get a weak result of this general type for groups with r.b. (p2 p 1). For smaller representation bounds, more information can be obtained. Our main result is:

THEOREM. Let G have r.b. (2p 8) for a prime p. Then either

(i) An Sp of G is normal and abelian,

(ii) G is solvable and has p-length 1 or

(iii) G = P ×H where P is an abelian p-group and p2 ↑|H|.

Mathematical Subject Classification
Primary: 20.25
Received: 25 March 1966
Published: 1 November 1967
I. Martin (Irving) Isaacs