College Algebra Exam Review 78

College Algebra Exam Review 78 - 88 2. BASIC THEORY OF...

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88 2. BASIC THEORY OF GROUPS The statement (c) means, for example, that any group of order 3 is isomorphic to Z 3 . Statement (d) means that there are two distinct (noniso- morphic) groups of order 4, and any group of order 4 must be isomorphic to one of them. Proof. The reader is guided through the proof of statements (a) through (e) in the exercises. The idea is to try to write down the group multiplication table, observing the constraint that each group element must appear exactly once in each row and column. Statements (a)–(e) give us the complete list, up to isomorphism, of groups of order no more than 5, and we can see by going through the list that all of them are abelian. Finally, we have already seen that Z 6 and S 3 are two nonisomorphic groups of order 6, and S 3 is nonabelian. n The deﬁnition of isomorphism says that under the bijection, the multi- plication tables of the two groups match up, so the two groups differ only by a renaming of elements. Since the multiplication tables match up, one
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This note was uploaded on 12/15/2011 for the course MAC 1105 taught by Professor Everage during the Fall '08 term at FSU.

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