2.10 The Isomorphism Theorem

1. Define img by img This is a group homomorphism by direct calculation. We have img if and only if a = 1 = c, so ker α = K.
3. Since img, H G and G/H = {H, GH} is a group with H as unity. Thus σ : G/H → {1, − 1} is an isomorphism if σ(H) = 1 and σ(GH) = − 1. Let ϕ : GG/H be the coset map ϕ(g) = Hg, and let α = σϕ : G → {1, − 1}. Then α is a homomorphism and: If g img H, then α(g) = σϕ(g) = σ(H) = 1; if gH, then α(g) = σϕ(g) = σ(GH) = − 1.
4.
a. We have 1 img α−1(X) because α(1) = 1 img X. If g, h img α−1(X), then α(g) img X and α(h) X, so α(g−1) = [α(g)]−1 X and α(gh) =

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