Groups whose elements have order 2

[halvizo1031]

Homework Statement

suppose that G is a group in which every non-identity element has order two. Show that G is commutative.

Homework Equations

The Attempt at a Solution

Is my answer correct?

Suppose that a,b and ab all have order two. we will show that a and b commute. By assumption, e=(ab)^2
=abab
As a and b are their own inverses, multiplying on the left by a and then b,
we get
ba=ab.

 

[VeeEight]

Looks good to me.
e = abab implies
b = ababb = aba which implies
ba = abaa = ab

 

[Dick]

Yes, that's right.

 

[halvizo1031]

Looks good to me.
e = abab implies
b = ababb = aba which implies
ba = abaa = ab

Thanks! Is there another way i can show this?

 

[Dick]

Thanks! Is there another way i can show this?

Why do you need another way? What did you have in mind?

 

[halvizo1031]

Why do you need another way? What did you have in mind?

well i just don't want to have the same answer as someone else.

 

[Dick]

well i just don't want to have the same answer as someone else.

It's a simple problem. There's a simple answer. There's a few different permutations on the expression of that answer, but they are all really the same. Wouldn't it be better to move on to the next problem?

 

[halvizo1031]

It's a simple problem. There's a simple answer. There's a few different permutations on the expression of that answer, but they are all really the same. Wouldn't it be better to move on to the next problem?

That's true. I'm having trouble showing that an element k is a generator of Zn if and only if k and n are relatively prime.

 

[Dick]

That's true. I'm having trouble showing that an element k is a generator of Zn if and only if k and n are relatively prime.

The order 2 part of the problem has nothing to do with proving k is a generator of Zn if k and n are relatively prime. Try thinking about them independently.

 

[halvizo1031]

The order 2 part of the problem has nothing to do with proving k is a generator of Zn if k and n are relatively prime. Try thinking about them independently.

Sorry, i forgot to mention that this question has nothing to do with the order 2 problem. Here's the new question:

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