Proving the Identity Property in Abelian Groups

[dorin1993]

Hi guys,

I have quastion about groups:

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!

 

[hedipaldi]

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

 

[micromass]

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

 

[dorin1993]

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

 

[HallsofIvy]

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

 

[dorin1993]

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

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)

 

[HallsofIvy]

Yes, that is what he meant.

 

[dorin1993]

Thank you so much! :)