Unformatted text preview: .˛ a ± 1 .n ± 1 /;a ± 1 / is the inverse of .n;a/ . 3.2.2. Show that j W ŒxŁ 7! Œ ± xŁ deﬁnes an order 2 automorphism Z n . Conclude that ˛ W Œ1Ł 2 7! j determines a homomorphism of Z 2 into Aut . Z n / . Prove that Z n Ì ˛ Z 2 is isomorphic to D n . 3.2.3. Show that the afﬁne group Aff .n/ is isomorphic to a semidirect product of GL .n; R / and the additive group R n . 3.2.4. Show that the permutation group S n is a semidirect product of Z 2 and the group of even permutations A n . 3.2.5. Consider the set G of n-by-n matrices with entries in f 0; ˙ 1 g that have exactly one nonzero entry in each row and column. These are called signed permutation matrices. Show that G is a group, and that G is a semidirect product of S n and the group of diagonal matrices with entries...
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- Fall '08