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5) The subgroup (12) in S3 is not normal. (6) Any subgroup of G contained in Z(G) is normal in G. 3 Prove that SL (n, R) is a normal subgroup of GL (n, R). 4 Let ϕ : G → H be a group homomorphism. Prove that the kernel of ϕ is a normal subgroup of G. 5 Prove that if H is a subgroup of G and H has at most two cosets, then H is a normal subgroup. 6 The goal of this exercise is to prove that A5 has no normal subgroups other than itself and the trivial subgroup. (1) Prove that any two 3-cycles in A5 are conjugate in A5 .

4 A principal ideal domain is a commutative domain with identity in which each ideal is generated by at most one element. We have proved that Z is a principal ideal domain. 5 Let F be a field and X a variable over F . Then the polynomial ring F [X] is a principal ideal domain. Proof. Let I be an ideal in F [X]. If I is (0), then done. So suppose that I contains a non-zero element. Let g be a non-zero element in I of least possible degree (as a polynomial in X). Claim: I = (g). Certainly (g) ⊆ I.

Vr and the sequence obtained from this one by switching vk and vk+1 give the same configuration. Now let the sequences v1 , v2 , . . , vr and w1 , . . , ws produce stable configurations, and let v1 = v2 = · · · = vi = v, w1 = w2 = · · · = wj = w. Since w is overfull, there exists k ∈ {i + 1, . . , r} such that w = vk . Let k be the smallest such integer. By repeated use of the end of the previous paragraph, v1 , . . , vr gives the same configuration as w = vk , v1 , . . , vk−1 , vk+1 , .

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