HOMEWORK 8 — DUE NOV 14TH

MATH 25

There are two sections: A and B. Please return each part to the proper CA.

A. Problems graded by Benjamin

A.1. In each of the following exercises, you are given a set G and a composition law (x, y) ↦ x ⊗ y. Determine whether (G, ⊗) is a group and if it is, find the unit element, give a formula for the inverse of an element, and determine whether G is abelian.

(1) G = R and x ⊗ y = x + y + xy.

(2) G = (R \ {0}) × R and (x, u) ⊗ (y, v) = (xy, xv + u).

(3) G = {(x, y) ∈ R^{2}, x^{2} + y^{2} = 1} and (x_{1}, y_{1}) ⊗ (x_{2}, y_{2}) = (x_{1}x_{2} − y_{1}y_{2}, x_{1}y_{2} + y_{1}x_{2}).

A.2. Let G be a finite group and let a, b ∈ G be such that (ab)^{m} = e for some m ≥ 1. Prove that (ba)^{m} = e.

A.3. Let G be a group such that for every a ∈ G, we have a^{2} = e. Prove that G is abelian.

A.4.

(1) Find all groups G which have no non-trivial automorphisms.

(2) Find all groups G which have no non-trivial subgroups.

A.5. Find all continuous homomorphisms f : (R, +) → (R, +).

B. Problems graded by Inna

B.1. Prove that if G is an infinite group, then it has an infinite number of subgroups.

B.2. Prove that if H is a subgroup of index 2 of a group G, then H is a normal subgroup of G.

2 MATH 25

B.3. The center Z = Z(G) of a group G is the set of elements z ∈ G such that zg = gz for every g ∈ G.

(1) Prove that Z is a subgroup of G.

(2) Prove that if G/Z is cyclic, then G is abelian (and so G = Z).

(3) Prove that if G is not abelian, then [G : Z] ≥ 4.

(4) Define C(G) = {(x, y) ∈ G^{2}, xy = yx}. Prove that if G is finite and not abelian,

then

#C(G) ≤ \frac{5}{8}(#G)^{2}.

B.4. Let S_{4} = Bij({a, b, c, d}). Prove that S_{4} has a unique subgroup of order 12. Call it A_{4}. Prove that A_{4} has no subgroup of order 6.

B.5. Prove that if H is a subgroup of a finite group G (and H ≠ G), then G ≠ ∪_{g∈G}(gHg^{−1}).