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### Homework 2

- Section 6.1
- Homework: 6, 7, 8, 9, 10, (Extension: 16, 22, 24, 25) and these problems:
- 39. Let \(G\) be a group. Show that if \(G/Z(G)\) is cyclic, then \(G\) is abelian. Hence show that if \(G\) has order \(p^2\) (where \(p\) is prime), then \(G\) is abelian.
- 40. If G is a nilpotent group of class 2, show
- G' ≤ Z(G), and
- [x
^{n},y] = [x,y]^{n}= [x,y^{n}] for all x, y ∈ G and all positive integers n.

- Hand In: 39, 40

- Homework: 6, 7, 8, 9, 10, (Extension: 16, 22, 24, 25) and these problems:
- Section 4.5
- Homework: 19, 23, 25, 30, 32 and these problems:
- 57. Prove that if \(\vert G\vert = 616\) then \(G\) is not simple.
- 58. Let \(P\) be a Sylow \(p\)-group of a finite group \(G\) and let \(H\) be a subgroup of \(G\) containing \(N_G(P)\). Prove that \(H = N_G(H)\). (Source: Derek J.S. Robinson, "Abstract Algebra (Second Edition)", De Gruyter, Berlin/Boston (2015))

- Hand In: 57, 58

- Homework: 19, 23, 25, 30, 32 and these problems:
- Section 6.2
- Homework: 6, 7, 11, 13, 14, 24 and these problems:
- 31. Prove that if \(\vert G\vert = 2376\) then \(G\) is not simple.
- 32. Prove that if \(\vert G\vert = 8000\) then \(G\) is not simple.

- Hand In: 31, 32

- Homework: 6, 7, 11, 13, 14, 24 and these problems:
- Section 3.4
- Homework: these problems:
- 13. Find the subgroup lattice of the group \(\mathbb{Z}_{72}\).
- 14. Given the two subnormal series \(\{0\} \leq \langle 4\rangle \leq \mathbb{Z}_{72}\) and \(\{0\} \leq \langle 18\rangle \leq \langle 6 \rangle \leq \mathbb{Z}_{72}\), follow the proof of Schreier's Theorem to find refinements of each series and exhibit the isomorphic factor groups, precisely as in the proof (including the trivial factors).

- Hand In: 13, 14

- Homework: these problems:
- Section 5.1
- Homework: 1, 5, 12, 14, 15, 17 and these problems:
- 19. Give an example of an abelian group \(H \times K\) that contains a nontrivial subgroup \(N\) such that \(N\cap H = 1 = N \cap K\). Conclude that it is possible that \(N \le H \times K\) and \(N \neq (N \cap H) \times (N \cap K)\). (Source: Joseph J. Rotman, "An Introduction to the Theory of Groups (Fourth Edition)", Springer, New York (1995))
- 20. Let \(G\) be a group having a simple subgroup \(H\) of index \(2\). Prove that either \(H\) is the unique proper normal subgroup of \(G\) or that \(G\) contains a normal subgroup \(K\) of order \(2\) with \(G = H \times K\). (Source: Joseph J. Rotman, "An Introduction to the Theory of Groups (Fourth Edition)", Springer, New York (1995))

- Hand In: 19, 20

- Homework: 1, 5, 12, 14, 15, 17 and these problems: