**Unformatted text preview: **Sample Questions for Midterm 1, MATH 435
Problem 1 [Review the deﬁnitions and theorems covered so far. You will be asked to state a few of them precisely.] Problem 2 [Review the TRUE/FALSE questions of the relevant sections.] Problem 3 Let G be a group. An automorphism is an isomorphism f : G → G. Let Aut(G) be the set of all automorphisms of G. Show that Aut(G) with the binary operation of composition is a subgroup of S G , the symmetric group of G. Problem 4 Let A be the square below. σ
900 τ 2 1 4 3 Let σ be the symmetry of the square given by counter-clockwise rotation by 90◦ . Let τ be the symmetry given by reﬂection along the diagonal through nodes 2 and 4. 1. Which permutations in S 4 are represented by σ, τ, respectively? Give a representation as (products of disjoint) cycles. 2. What is the index of the subgroup H of S 4 generated by {σ, τ}? 3. Is H cyclic? Problem 5 Let G be a group. Let a ∈ G. Deﬁne the mapping fa : G → G by fa ( x) = ax for all x ∈ G. Show that fa is an element of the symmetric group S G . ...

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