200a-f11-hw5 - 1 Let K and H be groups and let φ 1 K →...

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Math 200a Fall 2011 Homework 4 Due Friday 10/28/2011 by 5pm in homework box on 6th floor of AP&M Upcoming schedule: We will have a midterm exam on Wed. Nov. 2 in class. There will be no homework due on Friday of that week (Nov. 4). There will be a homework due the next week (Nov. 11) even though we have no class that day. Reading assignment: Beginning of 5.4, 6.1. We will not cover 6.2. Exercises to be handed in: (all exercise numbers refer to Dummit and Foote, 3rd edition.) Section 5.5: 2 , 12 Remark: Do additional exercise 1 below before doing exercise 5.5 #12 above. In 5.5 #12, I do want you to show that there are exactly 5 distinct groups up to isomorphism. Use additional exercise 1 to help you prove that some of the semi-direct products you get in 5.5 #12 are isomorphic to each other; this will allow you to show there are at most 5 distinct groups of order 20. You still need to check that the 5 you found are non-isomorphic. Section 6.1: 4 , 9 , 10 . Exercises not from the text: (all to be handed in):
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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 semi-direct 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 | ( q-1), 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 n-1 E G n = G such that for all 1 ≤ i ≤ n , G i /G i-1 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...
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