Mth 171 abstract algebra fall 2014- assignment 1 let g a

Question

Math 171: Abstract Algebra, Fall 2014- Assignment 1

1. Let G = {a + b√5: a, b ∈ Q}. Prove that the non-zero elements of G form a group under multiplication.

2. Let G = {x ∈ R: 0 ≤ x < 1} and for x, y ∈ G, define x * y to be the fractional part of x + y. Prove that * is a binary operation on G and that G is an abelian group under the operation *. (A group G is abelian of x * y = y * x for all x, y ∈ G).

3. Let n ≥ 2 be an integer. Consider the set S = {1, 2, 3, . . . , n}. Let G be the set of all bijections f: S → S, and * be composition of functions. Prove (G, *) is a group with n! elements in it.

4. Determine, with justification, whether each of the following statements is true or false in any group G.

(a) For n ≥ 2, (g^-1)ⁿ = (gⁿ)^-1 for all g ∈ G. Here, hⁿ:=

(b) (g * h)² = (h * g)² for all g, h ∈ G.

5. Let G be a group with |G| = 4. Prove G is abelian (i.e. commutative).