# Free Group with defining equations

1. Sep 13, 2011

### iamalexalright

So today in class we talked about defining equations...

We were asked to consider the group generated by <a,b>

with the defining equations a^2 = e, b^3 = e, and ba = ab^2. With these equations we can easily see that there can only be a maximum of 6 elements (and apparently most of the time there will be exactly six).

My professor recalled there was some group(s), defined in a similar fashion, that should have n elements but surprisingly has less.

Anybody have any insight on this? Sorry if this is too vague

2. Sep 14, 2011

### rasmhop

I can obviously not be sure what example your professor had in mind, but I recall an example that may be of a similar spirit (this is from Dummit-Foote if I recall correctly).
$$\langle x,y|x^n=y^2=e,\quad xy=yx^2\rangle$$
Here one may guess that this group has order 2n, but
$$x = xy^2 = yx^2y =yxyx^2= y^2x^4 = x^4$$
so $x^3 =e$ and therefore the group has at most order 6.

This is not especially surprising to more experienced mathematicians, but to an introductory abstract algebra student it may be surprising. Your professor may have had a more sophisticated example in mind which even baffles experienced mathematicians.