Let G be a cyclic group generated by an element a. a) Prove that if an...

Let G be a cyclic group generated by an element a.

a) Prove that if aⁿ = e for some n ∈ Z, then G is finite.

b) Prove that if G is an infinite cyclic group then it contains no nontrivial finite subgroups. (Hint: use part (a))

Expert Solution

Colby Messinger answered 2 years ago

Let G be a group and a be an element of G. Let φ:Z→G be a...

Let G be a group and a be an element of G. Let φ:Z→G be a map defined by φ(n) =a^{n} for all n∈Z. a)Show that φ is a group homomorphism. b) Find the image ofφ, i.e.φ(Z), and prove that it is a subgroup ofG.

Let G be a group, and let a ∈ G be a fixed element. Define a...

Let G be a group, and let a ∈ G be a fixed element. Define a function Φ : G → G by Φ(x) = ax−1a−1. Prove that Φ is an isomorphism is and only if the group G is abelian.

Let F be a finite field. Prove that the multiplicative group F*,x) is cyclic.

Let a be an element of a finite group G. The order of a is the...

Let a be an element of a finite group G. The order of a is the least power k such that ak = e. Find the orders of following elements in S5 a. (1 2 3 ) b. (1 3 2 4) c. (2 3) (1 4) d. (1 2) (3 5 4)

Let (G,+) be an abelian group and U a subgroup of G. Prove that G is...

Let (G,+) be an abelian group and U a subgroup of G. Prove that G is the direct product of U and V (where V a subgroup of G) if only if there is a homomorphism f : G → U with f|U = IdU

Let G be a group. For each x ∈ G and a,b ∈ Z+ a) prove...

Let G be a group. For each x ∈ G and a,b ∈ Z+ a) prove that xa+b = xaxb b) prove that (xa)-1 = x-a c) establish part a) for arbitrary integers a and b in Z (positive, negative or zero)