This month we will venture into Group Theory. Recall that a group is a set G with a binary operation * satisfying the following properties:
Note that in general it is not the case that a*b = b*a. From now on, we will suppress the * and just juxtapose elements.
Consider the group generated by a,b,c, and d subject to the relations
Using the first relation, the second relations becomes bab = d. Using this expression and the first relation, we obtain
Taking the second relation above and multiplying both sides on the left by a^{-1}b^{-1} and on the right by a^{-1}, we have b = a^{-2}. Now first relation above becomes aa^{-2}a^{-2}aa^{-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?
Source: John H. Conway
No correct solutions have been submitted, so this problem is still open.
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