# 41 is the Largest Size of a Cap in PG(4,4) by Edel Y., Bierbrauer J. By Edel Y., Bierbrauer J.

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Let us denote by 0(4) the group of all isometries in R 4 and by S3 := ( i e l 4 : \x\ = 1}. 2. Iff,g£ 0(4) satisfy ^(f + g) G 0(4), then fg~l = -gf~l. Proof. ))_1 = ^ I T ^ S - 1 ) . Hence, id = \{2i& +fg~l + S / " 1 ) , and then fg'1 + gf~l = 0 . • In the following lemma, we shall denote by Lin(S') the linear span of a set S c i . 3. Let f € 0(4) be satisfying f~l = —f. Let u,v £ S3 such that (u | v) = 0 and f(u) = v, and let be s,t s Lin(u, v)-1 satisfying s,t £ S 3 and (s | t) = 0. Then f(v) = —u, f(s) = et and f(t) = — es, where e = ± 1 .

We also give in the same reference an approach to the four-dimensional case as a consequence of the ideas in . The aim of this paper is to develop intrinsic techniques to the geometry of the quaternions to obtain a classification in the fourdimensional case. 2. s. we will identify T with HI as euclidean vector spaces. From now on, given x, y s H, the juxtaposition xy will mean the usual product in H. Let us introduce some terminology. Definition 2 . 1 . Let /J 6 S3 be a permutation of the set {1, 2,3}.

Indeed, let T be cogenerated by the Priifer group Zp0o for some prime p. Then the Z-module Z is clearly extending. It is easy to see that the natural homomorphism from the closed Z-submodule pL of Z to Z cannot be extended to an endomorphism of Z, hence Z is not a self-T-divisible Z-module. 33 In what follows, we shall be interested in when a finite direct sum of self-r-divisible modules is self-r-divisible. Towards such a result we give some preliminary technical properties. 3. Let A\ and A2 be modules, A = Ax © A2 and let pi, p2 be the canonical projections on A\ and A2 respectively.