Unformatted text preview: 1 . Let K and H be groups, and let φ 1 : K → Aut( H ) be a homorphism. Let G 1 = H o φ 1 K be the semidirect product. Suppose that σ : K → K is an automorphism of K . Then φ 2 = φ 1 ◦ σ : K → Aut( H ) is also a homomorphism, and so we can also consider the semidirect product G 2 = H o φ 2 K . (a). Prove that the groups G 1 and G 2 are isomorphic. (b). Use part (a) to justify the claim made in class that if with p < q are primes with p  ( q1), then there are precisely two groups of order pq up to isomorphism. 2 . A group G is called polycyclic if G has a series G = { e } E G 1 E ··· E G n1 E G n = G such that for all 1 ≤ i ≤ n , G i /G i1 is a cyclic group (ﬁnite or inﬁnite). (a). Show that if G is polycyclic, then any subgroup or factor group of G is again polycyclic. (b). Show that if N E G and G/N and N are polycyclic groups, then G is also polycyclic. 1...
View
Full Document
 Fall '08
 Bunch,J
 Math, Group Theory, Cyclic group, additional exercise, gn, semidirect product

Click to edit the document details