3.2. SEMIDIRECT PRODUCTS 155 that ' W x 7! .' 1 .x/;:::;' r .x// is an injective homomorphism into A 1 ± ²²² ± A r . (b) Explore conditions for ' to be surjective. 3.1.19. Let N 1 ;N 2 ;:::;N r be normal subgroups of a group G . Show that N i \ .N 1 :::N i ± 1 N i C 1 :::N r / D f e g for all i if, and only if, whenever x i 2 N i for 1 ³ i ³ r and x 1 x 2 ²²² x r D e , then x 1 D x 2 D ²²² D x r D e . 3.2. Semidirect Products We now consider a slightly more complicated way in which two groups can be ﬁt together to form a larger group. Example 3.2.1. Consider the dihedral group D n of order 2n , the rotation group of the regular n-gon. D n has a normal subgroup N of index 2 con-sisting of rotations about the axis through the centroid of the faces of the n-gon. N is cyclic of order n , generated by the rotation through an angle of 2±=n . D n also has a subgroup A of order 2, generated by a rotation j through an angle ± about an axis through the centers of opposite edges (if n is even), or through a vertex and the opposite edge (if
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