One knows that any countable group can be embedded in a 2-generated group. Also, any finite group is a subgroup of a 2-generated finite group, namely by the Cayley embedding. Does the same hold if we restrict to finite $p$-groups, where $p$ is some prime, i.e. is any finite $p$-group a subgroup of a 2-generated finite $p$-group?

Note that the minimal number of generators of the Sylow subgroup of the symmetric groups grow with their order. Thus the Cayley embedding would not suffice.


This is proved in

Neumann, B. H. and Neumann, H. (1959), Embedding Theorems for Groups. Journal of the London Mathematical Society, s1-34: 465–479. doi:10.1112/jlms/s1-34.4.465

At least, it says in the introduction that they prove this. For some reason I don't seem to be able to access the full article online at the moment.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.