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A 1.5-dimensional version of Hopfs Theorem on the number of by Bieri R. PDF

By Bieri R.

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A certain section C(t) This implies the existence of V in G Abelian p-group and on which which is an elementary L l acts non-trivially. The degrees of the irreducible Brauer characters of for the prime p being known, we find a lower bound for the dimension of V as a vector 35 space over the field of p elements. On the other hand, there is a four-group acting on V and we can find an upper bound for the dimension. The two bounds are compatible only when q2 = 3. Then V L l 5 , is the Frattini factor group of 3 an extra-special group of order action of = ql on V 5 , and the corresponds to an embedding in such a way that an element of order 3 leaves the vectors of a hyperbolic plane fixed.

The concept of an orbital is due to D. G. Higman [5]. orbit 6 (a) G 6: 0. l a has (G) i r th Definition 2 E Let mutation group relative to G E E " an A transitive group @ orbitals; see also Sims [8]). l orbital character I(a a in such a way that @ for all (aG) assigns to " n I aG E ~. (a)} 11:= 91 ~ l I by (G E @) be subgroups of the per- Define the support of ~ U and define a corresponding numerical relative support function s(91:U) 1~(91:U) I . 58 FINITE GROUPS The concept of an orbital character directly generalizes the notion of the permutation character _ ~ if we set -1 = e.

Despite the specific nature and simplicity of the condition 2p ~ 2 m + 1 , the only known examples of uniprimitive groups of degree p 6 a prime, occur when 5 p ~ 5 acting on the 2-subsets of (1, m ~ 3 2p, ~5 and 2, 3, 4, 5) ). Manning * and refers to primitive permutation groups which are not doubly transitive. The results announced in this article are part of the author's doctoral dissertation submitted to the faculty of Yale University. 55 56 FINITE GROUPS For a given prime p, the existence of a group which is uniprimitive of degree 2p implies the existence of a simple group with the same property.

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