Algebra I by Randall R. Holmes

By Randall R. Holmes

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Then for any aN ∈ G/N we have ψ(aN ) = ψ(π(a)) = (ψπ)(a) = ϕ(a) = ϕ(aN ), so that ψ = ϕ, thus establishing uniqueness. The condition N ⊆ ker ϕ is a necessary condition for the function ϕ given in the theorem to be well defined, as the reader can check. 8 – Exercises 1 a | a ∈ R viewed as a group under matrix multipli0 1 cation. Prove that G ∼ = R. 8–1 Let G = 46 8–2 Let G and G be groups, let ϕ : G → G be an epimorphism, let N be a normal subgroup of G, and assume that N ∩ ker ϕ = {e}. Prove that ϕ(CG (x)) = CG (ϕ(x)) for all x ∈ N .

Sm , T1 , . . , Tn . 10–4 Determine the composition factors of the group D8 × Z12 . 1 Definition Let n be a positive integer. 7) that Sn denotes the symmetric group of degree n, which is the group of all permutations of the set {1, 2, . . , n}, with binary operation being function composition (written using juxtaposition). An element σ of Sn has a 2 × n matrix representation σ= 1 2 ··· σ(1) σ(2) · · · n . σ(n) Since each such representation is completely determined by its bottom row (and different bottom rows correspond to different permutations), the number of permutations of {1, 2, .

I) For aN, bN ∈ G/N we have (aN )(bN ) = abN ∈ G/N . In particular, this product defines a binary operation · on G/N . (ii) (G/N, · ) is a group. Its identity element is eN (= N ) and for aN ∈ G/N we have (aN )−1 = a−1 N . Proof. We first observe that N N = N . Indeed N N ⊆ N by closure and, if n ∈ N , then n = ne ∈ N N giving the other inclusion as well. If aN, bN ∈ G/N , then (aN )(bN ) = abN b N = abN N = abN using normality of N . This gives part (i). Associativity of coset multiplication now follows immediately from the associativity of multiplication in G.

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