I first saw this on the Missouri State problem page and it has never been solved there.
Consider the group generated by a,b,c, and d subject to the relations
ab = c, bc = d, cd = a, and da = b
Using the first relation, the second relation becomes bab = d. Using this expression and the first relation, we obtain
abbab = a and baba = b
Taking the second relation above and multiplying both sides on the left by a^-1b^-1 and on the right by a^-1, we have b = a^-2. Now first relation above becomes aa^-2a^-2aa^-2 = a or a^-4 = a, hence i = a^5. Therefore our group consists of the five elements i, a, a^2, a^3, a^4. The other elements can be expressed in terms of a as follows: b = a^3, c = a^4, and d = a^2.
Finally, we get to his month's problems. How many elements are there in the groups given by the following generators and relations?
* Generators:a,b,c Relations: ab = c, bc = a, ca = b * Generators:a,b,c,d,e Relations: ab = c, bc = d, cd = e, de = a, ea = b * Generators:a,b,c,d,e,f Relations: ab = c, bc = d, cd = e, de = f, ef = a, fa = b
Source: John H. Conway
I recognize the first one as the quaternions but have made no progress on the other two.