1. Nov 18, 2012

### dorin1993

Hi guys,

G is abelian group with an identity element "e".
If xx=e then x=e.

Is it true or false?

I was thinking and my feeling is that it's true but I just can't prove it.

I started with:

(*) ae=ea=a
(*) aa^-1 = a^-1 a = e
those from the definition of Group

and now the assuming: aa=e

then:

aa^-1 = e = aa
a=a^-1
==> a^-1 a = aa = e

that's all i got.
Is anyone can halp?

thank you!

2. Nov 18, 2012

### hedipaldi

what about the additive group of integers modulu 4?2 has order 2.

3. Nov 18, 2012

### micromass

Staff Emeritus
If G is finite, then you can prove that your result is true if and only if G has odd order.

4. Nov 19, 2012

### dorin1993

But the main argument is about ALL abelian group with xx=e

5. Nov 19, 2012

### HallsofIvy

Staff Emeritus
To disprove a general statement, you only need one counterexample.

6. Nov 19, 2012

### dorin1993

Do you mean grouo af all integers -
the identity element is 0
and for example 2 +(mod4) 2 = 0
although 2 ≠ 0
(I still trying to understand the modulo)

7. Nov 19, 2012

### HallsofIvy

Staff Emeritus
Yes, that is what he meant.

8. Nov 19, 2012

### dorin1993

Thank you so much! :)