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.

up vote 14 down vote accepted

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.

  • How about if those two generators come from a well behaved group like $SL(2, F_q)$? – user76098 Jul 27 '15 at 18:31
  • 4
    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
  • 1
    Oops: `...every finite group embeds in a finite 2-generated group.' – shane.orourke Jul 28 '15 at 6:44
  • 1
    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. – Richard Rast Jul 28 '15 at 12:04
  • 1
    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

Your Answer

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.