Introduction to Abstract Algebra, Solutions Manual, 4th Edition

2.5 Homomorphisms and Isomorphisms

1. a. It is a homomorphism because

It is clearly not onto, but it is one-to-one because α(r) = α(s) implies so r = s.

3. If α is an automorphism, then a⁻¹b⁻¹ = α(a) · α(b) = α(ab) = (ab)⁻¹ = b⁻¹a⁻¹ for all a, b. Thus G is abelian. Conversely, if G is abelian,

so α is a homomorphism; α is a bijection because α⁻¹ = α.

5. σ_a = 1_G if and only if aga⁻¹ = g for all g R, if and only if ag = ga for all g R.

7. Let be given by . Then o(1) =∞ in , but in .