Groups whose order have order two

[halvizo1031]

I need help here: Suppose that G is a group in which every non-identity element has order two. Show that G is commutative.


Also, Consider Zn = {0,1,...,n-1}
a. show that an element k is a generator of Zn if and only if k and n are relatively prime.

b. Is every subgroup of Zn cyclic? If so, give a proof. If not, provide an example.

 

[VeeEight]

The forum rules state that you must show your attempt at a solution.

 

[halvizo1031]

The forum rules state that you must show your attempt at a solution.

I would show an attempt if I knew how to start it.

 

[VeeEight]

Here are some hints:
a) If k generates Zn, then k must have order n. But the order of <k> = n / gcd(k,n).
b) Are subgroups of cyclic groups necessarily cyclic?

 

[halvizo1031]

Here are some hints:
a) If k generates Zn, then k must have order n. But the order of <k> = n / gcd(k,n).
b) Are subgroups of cyclic groups necessarily cyclic?

b) as far as i know, subgroups of cyclic groups are always cyclic. but to be honest, I do not know if we are allowed to assume Zn is cyclic to begin with. It states that Zn = {0,1,...(n-1)}.

a) so because k generates Zn, generating k (x) amount of times will give us all the elements in Zn?

 

[Hurkyl]

I need help here: Suppose that G is a group in which every non-identity element has order two. Show that G is commutative.

I'm going to assume you've already spent a good amount of time experimenting with algebraic expressions to which you can apply x²=1 in creative ways.

If you haven't, you really should have.


So if the full question is too hard for you, then try a simpler problem first.

First, try to prove it in the case where G has zero generators.
Now, try to prove it in the case where G has one generators.
Now, try to prove it in the case where G has two generators.
Figure it out yet? No? Then try three generators...