## Math 5110 Graduate Algebra I

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### Homework 2

Due Friday September 28, in class
• 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
1. G' ≤ Z(G), and
2. [xn,y] = [x,y]n = [x,yn] for all x, y ∈ G and all positive integers n.
• Hand In: 39, 40
• 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
• 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
• 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
• 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