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Math 653 Homework Assignment 1 1. Let f : G → H be a group homomorphism. Prove the following: (a) f (eG ) = eH . (b) f (g −1 ) = f (g)−1 for all g ∈ G. (c) Kerf is a subgroup of G. (d) Imf is a subgroup of H. 2. Let G be a group such that g 2 = 1 for all g ∈ G. Prove that G is abelian. 3. Let G be a semigroup for which (i) there exists e ∈ G such that ae = a for all a ∈ G and (ii) for each a ∈ G there exists b ∈ G such that ab = e. Prove that G is a group. a b | a, b, c ∈ R, ac 6= 0}. Show that B2 (R) is a subgroup of 4. Let B2 (R) = { 0 c GL2 (R). (This subgroup is called the Borel group.) 5. The center of a group G is the set C(G) = {a ∈ G | ax = xa for all x ∈ G}. For a fixed g ∈ G, the centralizer of g is the set CG (g) = {a ∈ G | ag = ga}. Prove that C(G) and CG (g) are subgroups of G. 6. Let f : G → H be an injective group homomorphism. (a) Prove that |f (g)| = |g| for all g ∈ G. (b) Is the statement in (a) true in general if f is not injective? Prove or give a counterexample. a b × for all a, b ∈ R. 7. Let f : C → GL2 (R) be defined by f (a + bi) = −b a (a) Show that f is an injective group homomorphism. (b) Let S 1 denote the unit circle in the complex plane. Show that S 1 = f −1 (SL2 (R)). 8. Let G be a finite group having an automorphism σ (that is, an isomorphism from G to itself) such that σ(g) = g if and only if g = e. If σ 2 is the identity map, prove that G is abelian. (Hint: Show that every element can be written in the form g −1 σ(g) and apply σ to such an expression.) 9. (a) List the elements of the symmetric group S4 in cycle notation. (b) Prove that the dihedral group D24 (of order 24) and S4 are not isomorphic. 1