# Is there a structure theorem or group law for finite groups generated by two elements?

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.

• How about if those two generators come from a well behaved group like $SL(2, F_q)$? – user76098 Jul 27 '15 at 18:31
• The symmetric group on finitely many letters is generated by a long' cycle and a transposition; therefore every finite group embeds in a 2-generated group. – shane.orourke Jul 28 '15 at 6:36
• According to Theorem 2.1 of F. Levin, Factor Groups of the Modular Group, J. London Math. Soc 43 (1968), 195-203, every countable group is embeddable in a $2$-generator group with generators of prescribed orders $p\ge3$ and $q\ge2.$ – bof Aug 3 '15 at 10:07