By Salzmann H.

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**Example text**

Then N(W(S*» = S*, since S* is a maximal subgroup of G*. Thus N(W(S*» GLOBAL AND LOCAL PROPERTIES OF FINITE GROUPS has a normal p-complement. OP(G*) is a p'-group and 29 If G* has a normal p-complement, then [Op(G*), OP(G*)J £ Op(G*)nOP(G*)= 1, contrary to C(Op(G*» £ Op(G*). Hence G* does not have a normal pcomplement. Thus it suffices to assume that G* = G and that S* = S. For every subgroup H of G that contains Y, let H = HI Y. Let Eo be the setofallhESforwhich[X,h;p-l] = 1. SetE =

Let n be a positive integer. Suppose G S;; GL(n, p) and A is an abelian p-subgroup of G. Consider G to be a group of linear transformations on a vector space V of dimension n over GF(p). Let W be the subspace of V consisting of all the vectors fixed by every element of A. Assume that (g-l) (h-l) = 0 for all g, hE A and that G is generated by the conjugates of A in G. What are the possibilities for G and V if Op(G) = 1 and if either p is odd or [v/WI :s;; IAl? An important parti

Proof Since S £ N(T), we may assume that G = N(T). Let C = C(T). 10, C n S is a Sylow p-subgroup of C. Since C n S ~ T, Cn S = Z(T) £ Z(C). 2, OP(C) is ap'-group and OP(G) = OP(C). 1. 9. Digression Section &marks the end of the first part of this work. In the previous sections. we have discussed rather formal, general properties 'offinite groups. We have relied on only a few facts beyond the elementary theory of groups. 1 and some of its easy consequences, proved in Section 2. Let us recall the information files (A), (B), (C), and (D) mentioned in the introduction.

### 4-Dimensional projective planes of Lenz type III by Salzmann H.

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