Unformatted text preview: ± 1 / the inverse of .a;b/ . You are asked to check this in Exercise 3.1.1 . Deﬁnition 3.1.1. A ± B , with this group structure, is called the direct product of A and B . Example 3.1.2. Suppose we have two sets of objects, of different types, say ﬁve apples on one table and four bananas on another table. Let A denote the group of permutations of the apples, A Š S 5 , and let B denote the group of permutations of the bananas, B Š S 4 . The symmetries of the entire conﬁguration of fruit consist of permutations of the apples among themselves and of the bananas among themselves. That is, a symmetry is a pair of permutations .±;²/ , where ± 2 A and ² 2 B . Two such symmetries are composed by separately composing the symmetries of apples and the 147...
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 Fall '08
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 Algebra, Group Theory, quotient group G=N

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