Question

Let G be a group and suppose H = {g5 : g ∈ G} is a...

Let G be a group and suppose H = {g5 : g ∈ G} is a subgroup of G.

(a) Prove that H is normal subgroup of G.
(b) Prove that every element in G/H has order at most 5.

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Homework Answers

Answer #1

a) Consider

And is arbitrary

Then we have

This is because as the terms cancel out (the outer a inverse and inner a)

So that also

Meaning must be a normal subgroup

b) Let

Then we have so that is the identity element

This means the order of an element must be less than or equal to 5

That is, every element can have order at most 5

Hope this was helpful. Please do leave a positive rating if you liked this answer. Thanks and have a good day!

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