TITLE: Is there a structure theorem or group law for finite groups generated by two elements?
QUESTION [5 upvotes]: Say that $a, b \in G$ are two elements of a finite group $G$. Is there a structure theorem for the structure of $\langle a,b\rangle$? Is there a way to derive group laws for the group operation in the generated group?
I can think of special cases (the two elements commute, one of the elements is a power of the other, the commutator of the two elements commutes with them, etc.). I am wondering if a general classification results exists, still.

REPLY [15 votes]: It is a theorem of Graham Higman, Bernhard Neumann, and Hanna Neumann (Embedding theorems for groups, J. London Math. Society 24 (1949) 247-254) that every countable group can be embedded in a 2-generator group. This was later simplified by Bernhard and Hanna Neumann (Embedding theorems for groups, J. London Math. Soc. 34 (1959) 465-479). Fred Galvin has a paper in the Monthly (Embedding Countable Groups in 2-Generator Groups, Amer. Math. Monthly 100 no. 6 (1993), 578-580; available from JSTOR ) giving a simple proof, showing that in fact:

Theorem. Every countable group is embeddable in a 2-generator group, with one generator of order 11 and the other of order 2.

Given this, it seems hopeless to expect a structure theorem for 2-generator groups.