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College Algebra Exam Review 137

# College Algebra Exam Review 137 - ± 1 the inverse of.a;b...

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Chapter 3 Products of Groups 3.1. Direct Products Whenever we have a normal subgroup N of a group G , the group G is in some sense built up from N and the quotient group G=N , both of which are in general smaller and simpler than G itself. So a route to understanding G is to understand N and G=N , and ﬁnally to understand the way the two interact. The simplest way in which a group can be built up from two groups is without any interaction between the two; the resulting group is called the direct product of the two groups. As a set, the direct product of two groups A and B is the Cartesian product A ± B . We deﬁne a multiplication on A ± B by declaring the product .a;b/.a 0 ;b 0 / to be .aa 0 ;bb 0 / . That is, the multiplication is per- formed coordinate by coordinate. It is straightforward to check that this product makes A ± B a group, with e D .e A ;e B / serving as the identity element, and .a ± 1 ;b
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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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