By Mazurov V. D.

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5 Definition. A group G is called a split extension of a group H by a group F if H G and G contains a subgroup F1 such that F1 Š F , H \ F1 D f1g and HF1 D G. Alternatively one says that G is a semidirect product of H by F . The notation is G D H Ì F . Obviously, F Š G=H . 6 Examples. 1) Sn D An Ì Z2 for n > 2. 4). 7 Exercise. 5/=f˙Eg Š A5 , where E is the identity matrix. 5/ is not a split extension of Z2 by A5 . Hint. 5/ would not contain an element of order 4, which is not true. 8 Proposition.

Let G be an arbitrary extension of H by F . It is enough to prove that G contains a subgroup of order n. We prove this by induction on m. For m D 1 this is trivial, so let m > 1. We may assume that H is a proper subgroup of G. First consider the case where H contains a proper nontrivial subgroup H1 normal in G. H=H1 / Š F and by induction the group G=H1 contains a subgroup N=H1 of order n. Again by induction, N contains a subgroup of order n. Now consider the alternative case: H is a minimal nontrivial normal subgroup of G.

XN 1 / D 1; '. ux1 C x2 / D ux1 C x2 ; '. 1/ D xN 1 ; '. u C 1/x1 C vx2 C v 1x 3 where u 2 F4 , v 2 F4 n f0g. Obviously ' 2 is the identity mapping. The standard blocks passing through xN 1 (see Figure 2) are '-invariant, and ' carries the standard block passing through xN 2 and xN 3 to the oval O. Therefore ' can be informally considered as an inversion in M analogous to an inversion in the extended Euclidean plane. 4/. 16. , ' carries blocks of M to blocks of M . First we will verify that ' carries the standard blocks which do not pass through xN 1 to the nonstandard ones.