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5. SEMIDIRECT AND CENTRAL PRODUCTS If H

H(xjx) if i =f. j. xy for x, y in G. x is a homomorphism of G into the symmetric group Sn on the set S. H is the set of x in G fixing each Hx i . Equivalently x E K if and only if x E Xi- I HX i for all i, I ~ i ~ n. Thus 34 Some Basic Topics [Chap. 2] We have that G/ K, the image ofnH' is isomorphic to a group of permutations of S. Moreover, G/ K acts transitively on S, for if Hx;, HXj are two elements of S and we set x = Xi- 1 x j , then n x transforms HX j into Hx j . If we assume, as we may, that XI = 1, we verify direCtly that (H)nH is the subgroup fixing the letter HX I and more generally that (HXi)nH is the subgroup fixing the letter HXj of S.

Thus (w)xhx- I = 11' and consequently h fixes (IV)X for all h in H. Hence (w)x E Wand the lemma follows. 3 Let G be a p-group of linear transformations acting on a vector space V over a field F of characteristic p. Then some nonzero cector of V is fixed by every element of G. Proof We proceed by induction on IGI. Let M be a maximal subgroup of G. Then we have M

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4-dimensional compact projective planes with a 7-dimensional collineation group by Salzamann H. R.


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