Assignment 2

Name:.........................................................

1. a) For sets H and K, we know what H ∩K is, i.e. H ∩K = {x|x ∈ H and x ∈ K}. Showthat if H ≤ G and K ≤ G, then H ∩K ≤ G.

b) Let |H| = 12 and |K| = 35, find |H ∩K|.

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2. a) Let G be a cyclic group with generator a and let G′

be a group isomorphic to G. Ifφ : G→ G

′is an isomorphism, show that for every x ∈ G,φ(x) is completely determined by the

value φ(a), i.e. if φ : G → G′

and ψ : G → G′

are two isomorphisms such that φ(a) = ψ(a),then φ(x) = ψ(x) for all x ∈ G.

b) An isomorphism of a group with itself is called an automorphism of the group. Use theresult in a) to determine the number of automorphisms of Z12.

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3. Show that a group with no proper nontrivial subgroups is cyclic.

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4. a) Show that every finite group of even order 2n contains an element of order 2.

*b) Use the theorem of Lagrange to show that if n is odd, then an abelian group of order 2ncontains precisely one element of order 2.

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BONUS QUESTION *5. Let φ : G→ H be a group homomorphism. Show that φ(G) is abelianif and only if for all x, y ∈ G, xyx−1y−1 ∈ Ker(φ).

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