Math 405 - Fall 2016

Name: In-Class Exam II

1. (5 points each) Define the following.

(a) State Caley’s Theorem

(b) State the Fundamental Theorem of Finitely Generated Abelian Groups.

(c) A quotient group G/H.

(d) State the Fundamental Homomorphism Theorem.

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2. (3 points each) True or False: Circle one. You do not have to provide any explanation.

(a) T F LaGrange’s theorem states that every subgroup of a group has order that isa multiple of the order of the group.

(b) T F The set of all even permutations in S10 forms a group.

(c) T F The groups Z6 × Z10 and Z30 × Z2 are isomorphic.

(d) T F The group Z25 ⊕ Z4 ⊕ Z4 contains an element of order 400.

(e) T F For groups K ≤ H ≤ G with (H : K) and (G : H) both finite, we have

(G : K) = (G : H)/(H : K).

(f) T F The group Z×D3 is generated by the set {(1, ρ0), (0, µ1)}.

(g) T F If φ : G→ G′ is a group homomorphism, then Ker(φ) is a normal subgroupof G′.

(h) T F A nontrivial homomorphism exists from Z3 → Z.

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Short Computations: You must show all work to receive full credit.

3. For this problem, consider the following two permutations in S7.

α =

(1 2 3 4 5 6 74 3 2 5 7 1 6

), β =

(1 2 3 4 5 6 72 4 7 5 1 3 6

)(a) (5 points) Write α as a product of transpositions. Is α even or odd?

(b) (5 points) Compute βα and write it as a product of disjoint cycles.

4. (10 points) List all possible group structures, up to isomorphism, for abelian groups of order540.

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Computations: You must show all work to receive full credit.

5. (5 points) Find all left cosets of 20Z in 4Z.

6. (5 points) List the symmetries of the object below. Determine which well known group isisomorphic to the group of symmetries of the object.

7. (5 points) Let G = G1 × G2 × G3 be a direct product of groups with identity elements eirespectively. The second projection map π2 : G → G2 is defined as π2(g1, g2, g3) = g2.Determine the kernel of π2.

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8. (6 points) State fully and then write a proof synopsis of one of the following theorems. (Hint:It may be easiest to write down an actual proof of the statement on scratch paper and thenwrite your synopsis on the exam rather than directly creating the synopsis.)

(a) If H ≤ G of index 2 with G a finite group, then the left and right cosets of H agree.

(b) Every group of prime order is cyclic.

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