If we have a presentation $\langle x_i: R_j\rangle$ of $G$,

(1) what does the presentation (in terms of the given generators and the given relations) of a subgroup of $G$ looks like? In particular, what does the presentation of a normal subgroup $N$ looks like?

(2) What does the presentation of the quotient group $G/N$ looks like?

It seems that a subgroup should have less generators and the same relations, a quotient group should have same generators but more relations. Is this true?

  • $\begingroup$ Yes you can always find a presentation of a quotient group with the same generators and more relations. But a subgroup is more complicated, and in general it might need more generators, not less. There is a general method called the Reidemeister-Schreier algorithm for finding a presentation of a subgroup. $\endgroup$ – Derek Holt Oct 31 '16 at 19:27

(1) It depends. Removing some generators, and keeping all relations will give you a subgroup (you have to translate all relations involving the removed generators, you cannot simply remove them).

EDIT: Actually, "translating" the relations to a smaller set of generators won't work in general at all. Thanks to Derek.

Also, you can not create all subgroups in this way. A subgroup might in fact need more generators than the whole group. For example in the case without any relations (i.e. the free group), there is a nice (and possibly counter-intuitive) theorem that $\langle x_1,...,x_n:\emptyset\rangle$ is a subgroup of $\langle y_1, y_2 :\emptyset\rangle$ for any $k$.

(2) yes. Just add all relations of $N$ and you get the factor group $G/N$.

  • 1
    $\begingroup$ In your answer to (1), you may need to add extra relations to get a presentation of the subgroup generated by a subset of the generators. It is even possible for the original $R$ to be finite, but for the subgroup not to be finitely presented. $\endgroup$ – Derek Holt Oct 31 '16 at 20:20

If you want more detail for (2), there's a really good proof (which gives an idea of how to construct such a presentation) of presentations of factor groups on pages 42/43 of D.L.Johnson's book on presentations


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.