# ahw8 - Math 4124 Wednesday April 20 Eighth Homework...

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Math 4124 Wednesday, April 20 Eighth Homework Solutions 1. Exercise 5.2.1(e). Determine the number of nonisomorphic abelian groups of order 2704. First we write 2704 as a product of prime powers, namely 2 4 · 13 2 . To ﬁnd the number of abelian groups of order 16, we ﬁnd the number of partitions of 4. The partitions are (4), (3,1), (2,2), (2,1,1), (1,1,1,1) (so for example (3,1) corresponds to the group Z / 8 Z × Z / 2 Z ). Thus there are 5 abelian groups of order 16. Also there are 2 abelian groups of order 9. Therefore the total number of abelian groups of order 2704 is 5 · 2 = 10. 2. Exercises 5.2.2(e) and 5.2.3(e). List the elementary divisors and invariant factors for the abelian groups of order 44100. 44100 = 2 2 · 3 2 · 5 2 · 7 2 . Thus the number of abelian groups of order 44100 is 2 4 = 16. In the table below n will indicate Z / n Z ; thus using this notation there are two abelian groups of order p 2 , namely p 2 and p , p .

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ahw8 - Math 4124 Wednesday April 20 Eighth Homework...

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