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Abstract Algebra Practice Exam

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MATH 421/521 Section B Intro to Abstract Algebra HW2 — Fall 2020 All homework are required to be typed in LaTeX. You can use the free online editor http://www.overleaf.com. See https://www.overleaf.com/learn/ for a brief introduction. HW 2 is due Sunday September 6, by 11:59pm. Please upload your solutions on canvas under Assignments by the due time. 1. Let G be a group and H and F two different proper subgroups of G. (a) Prove that H ∩ F is also a subgroup of G. (b) Prove that if H and F do not contain each other then H ∪ F is not a subgroup of G. (hint: by our assumption there is an element a that lies in H but not in F and an element that lies in F but not in H. Suppose H ∪ F were a subgroup of G, derive a contradiction through a, b.) 2. (a) Let G be an abelian group and H = {x ∈ G : |x| is odd}. Prove that H is a subgroup of G. (b) Let G be a group and a an element of G. Prove that C(a) ⊆ C(an ) for every n ∈ Z+ , where C(x) denotes the centralizer of x in G. 3. Let m, n be elements of the additive group Z. Determine a generator (with justification) of hmi ∩ hni. 4. Let G be a group. Let p, q be two different prime numbers. Suppose a, b ∈ G are elements in G o ...
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