MIDTERM 2 FOR ALGEBRA
Date: 1999, November 29, 10:10–11:00AM
Each of Problems 1–5 is worth 14 points, each of problems 6-7 is worth 15 points.
1. Give an example of a group G such that |G| = 12 and G is not abelian.
2. Let G, H be two groups. Suppose that M is a normal subgroup of G and N is a normal subgroup of H. Show that M × N is a normal subgroup of G × H.
3. Find kernel of φ and φ(14) for φ : Z24→ S8 where φ(1) = (25)(1467).
4. Show that the commutator subgroup of Sn is contained in An. (Hint: consider the homomorphism φ : Sn→ Z2by φ(σ) = 1 if σ is odd and φ(σ) = 0 if σ is even.)
5. Find a composition series of S3× Z2.
6. Let G be the group hR, +i and X = R2. Let φ : G × X → X be defined by φ(t, (r cos θ, r sin θ)) = (r cos(θ + t), r sin(θ + t)).
Show that X is a G-set via the map φ. Let P = (1, 0) ∈ X. Find the isotropic subgroup GP. 7. Let K and L be normal subgroups of G with K ∨ L = G and K ∩ L = {e}. Show that G/K ' L.
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