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.



Is there a structure theorem or group law for finite groups generated by two elements?
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](http://www.jstor.org/stable/2324618) ) giving a simple proof, showing that in fact:



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> **Theorem.** Every countable group is embeddable in a 2-generator group, with one generator of order 11 and the other of order 2.
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Given this, it seems hopeless to expect a structure theorem for 2-generator groups. 


