College Algebra Exam Review 77

College Algebra Exam Review 77 - S 3 is nonabelian and Z 6...

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2.1. FIRST RESULTS 87 Recall that we have a notion of two groups being essentially the same: Definition 2.1.13. We say that two groups G and H are isomorphic if there is a bijection ' W G ±! H such that for all g 1 ;g 2 2 G '.g 1 g 2 / D '.g 1 /'.g 2 / . The map ' is called an isomorphism . You are asked to show in Exercise 2.1.5 that Z 4 is not isomorphic to the group of rotational symmetries of a rectangular card. Thus there exist at least two nonisomorphic groups of order 4. Definition 2.1.14. A group G is called abelian (or commutative ) if for all elements a;b 2 G , the products in the two orders are equal: ab D ba . Example 2.1.15. For any natural number n , the symmetric group S n is a group of order . According to Exercise 1.5.8 , for all n ² 3 , S n is nonabelian. If two groups are isomorphic, then either they are both abelian or both nonabelian. That is, if one of two groups is abelian and the other is non- abelian, then the two groups are not isomorphic (exercise). Example 2.1.16.
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Unformatted text preview: S 3 is nonabelian and Z 6 is abelian. Thus these are two nonisomorphic groups of order 6. So far we have one example each of groups of order 1, 2, 3, and 5 and two examples each of groups of order 4 and 6. In fact, we can classify all groups of order no more than 5: Proposition 2.1.17. (a) Up to isomorphism, Z 1 is the unique group of order 1. (b) Up to isomorphism, Z 2 is the unique group of order 2. (c) Up to isomorphism, Z 3 is the unique group of order 3. (d) Up to isomorphism, there are exactly two groups of order 4, namely Z 4 , and the group of rotational symmetries of the rectan-gular card. (e) Up to isomorphism, Z 5 is the unique group of order 5. (f) All groups of order no more than 5 are abelian. (g) There are at least two nonisomorphic groups of order 6, one abelian and one nonabelian....
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