# Thread: Presentation of a group in standard form

1. ## Presentation of a group in standard form

Hi

I have the following group presentation
G = a,b,c | ab = ba, ac = ca, bc = cb, a^2 = b^3 = c^3 = e.
I need to show that all the elements of G can be written in the standard form
a^ib^jc^k for i ∈{0,1} and j,k ∈{0,1,2}.
Thanks, Alex

2. ## Re: Presentation of a group in standard form

The fact that the given "presentation" starts "a, b, c" says that any member of G can be written as products of a, b, and c. The fact that "ab= ba, ac= ca, bc= cb" means that the order does not matter so any member is of the form "a^ib^jc^k" for some i, j, k. Use the conditions "a^2= b^3= c^3= e" to show that i need not be larger than 2 (fpr example a^5= (a^2)(a^2)(a)= (e)(e)(a)= a) and j and k need not be larger than 3.

3. ## Re: Presentation of a group in standard form

Thank you! That does seem to make sense.
Would it be enough to just give one example to show that i need not be larger than 2 and j,k need not be larger than 3, or would I need a general proof to show this?

4. ## Re: Presentation of a group in standard form

No. You need to use the fact that, for any i, either i= 2n or i= 2n+ 1 and that, for any j and k, j= 3m or 3m+1 or 3m+ 2 and k= 3p or 3p+ 1 or 3p+ 2.

5. ## Re: Presentation of a group in standard form

Ok so I have a^2n=(a^n)^2=e, and a^2n+1=a(a^n)^2=ae=a, hence i need not be larger than 2, and as a^2=e, we have i=0 or 1.
Is this correct? If so then I think I feel confident enough to do the same for b and c.

6. ## Re: Presentation of a group in standard form

Yes, that is true.