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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.

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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.

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  • $\begingroup$ How about if those two generators come from a well behaved group like $SL(2, F_q)$? $\endgroup$ – user76098 Jul 27 '15 at 18:31
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    $\begingroup$ 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. $\endgroup$ – shane.orourke Jul 28 '15 at 6:36
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    $\begingroup$ Oops: `...every finite group embeds in a finite 2-generated group.' $\endgroup$ – shane.orourke Jul 28 '15 at 6:44
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    $\begingroup$ Not to completely disagree with your heuristic, but every field embeds in an algebraically closed field, but the latter class is significantly easier to classify than the former. $\endgroup$ – Richard Rast Jul 28 '15 at 12:04
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    $\begingroup$ 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 $q\ge2.$ $\endgroup$ – bof Aug 3 '15 at 10:07

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