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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|.
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