Let be a prime and let be an abelian -group, where for each . Define the th-power map by .
- Prove that is a homomorphism.
- Describe the image and kernel of in terms of the generators .
- Prove that both and have rank (i.e. the same rank as and prove that these groups are both isomorphic to the elementary abelian group .
Without loss of generality, has type .
- Let be arbitrary. By this previous exercise, . Thus is a homomorphism.
- First we show that .
Suppose . Now for some , and , so that for each . Thus divides . We have for some , so that for each ; hence .
If , we have for some , for each . Then , so that , hence .
Now we show that .
Suppose . Then there exists with . Now for some for each , so that . Thus , hence .
Suppose . Then for each , there exists with . Clearly then , so that .
- We begin with some lemmas.
Lemma 1: Let be a cyclic -group of order , and let denote the -power map. Then . Proof: By the above argument, and . By this previous exercise, and , so that by Lagrange, . Now and are both cyclic, hence .
Lemma 2: Let be an abelian group, an integer, let denote the -power map, let , and let denote the -power map. Then . Proof: If , then .
Now to the main result.
Note that .
Now . By this previous exercise, , using Lemmas 1 and 2.
Thus . In particular, and have rank .