Let and be groups, let be a positive integer, let be a homomorphism, and let . In this previous exercise, we defined an injective homomorphism by permuting the factors. Now is a homomorphism . The wreath product of by via is defined . Note that the wreath product depends on the choice of permutation representation; if none is given, is assumed to be the left regular representation.
- Assume and are finite and is the left regular representation of . Find in terms of and .
- Let be a prime, let , and let be the left regular representation of . Prove that is a nonabelian group of order and is isomorphic to a Sylow -subgroup of . [The copies of whose direct product makes up may be represented by disjoint -cycles; these are cyclically permuted by .]
- We have . Thus .
- Since the left regular representation is nontrivial, and is nontrivial, is a nonabelian group. By part (1), this group has order .
Now is a finite abelian group. Let , , be disjoint -cycles in . By this previous exercise, there is a unique group homomorphism such that , where is the th “standard basis” vector. Moreover, . By our solution to this previous exercise, there exists a permutation such that , where indices are taken mod . Define by .
I claim that is a homomorphism. To see this, write and . Then .
Moreover, I claim that is injective; in this previous exercise we saw that every element of can be written uniquely as a product . Thus is injective. Counting elements, we see that is a Sylow -subgroup of .