# Are there a surface group and its two isomorphic subgroups which cannot be transferred to each other under any automorphism of the mother group? [closed]

My question is:

1. Are there $G$, which is a fundamental group of a surface or a 2-orbifal, and its two finite index subgroups $H_1$, $H_2$ satisfying the following condition: $H_1$ is isomorphic to $H_2$, but for any automorphism $\varphi$ of $G$, $\varphi(H_1)$ is not $H_2$?

2. If the answer to question 1 is Yes, can $H_1$ and $H_2$ be normal subgroups of $G$?

This question arise from the extension of the automorphism of a surface or an orbifal to its covering space.

-
Yes to all questions: just take $G$ equals the integers and $H_1$, $H_2$ be the subgroups generated by $p$ and $q$ with $|p|$ different from $|q|$. All the groups involved are infinite cyclic and hence isomorphic to the fundamental group of the annulus. –  Bruno Martelli Oct 1 '12 at 12:39
Do you really mean 'homeomorphic'? Isomorphic' would seem to make more sense (and the answer to 1 and 2 would then be YES.) If you really mean 'homeomorphic' you'd better give us more information about what this group G is, e.g. it is, at least, a topological group. –  Nick Gill Oct 1 '12 at 12:42
This question doesn't look like a research level question to me. –  Fernando Muro Oct 1 '12 at 13:18
Sorry for the first two stupid questions. I originally think of the covering space (hence the subgroup) of some surfaces. May the title should be "Are there two homeomorphic subgroups of a surface group which cannot be transferred to each other under any automorphism of the mother group?" –  X.M. Du Oct 1 '12 at 14:54
@X.M. Du Please edit your question so that it asks what you want in a clear and precise way. The guidelines here are useful mathoverflow.net/howtoask . Then it has a chance of being reopened. –  j.c. Oct 1 '12 at 17:09
show 4 more comments

## closed as too localized by Fernando Muro, Mark Sapir, Anton Petrunin, Benoît Kloeckner, Igor RivinOct 1 '12 at 15:02

This question is unlikely to help any future visitors; it is only relevant to a small geographic area, a specific moment in time, or an extraordinarily narrow situation that is not generally applicable to the worldwide audience of the internet. For help making this question more broadly applicable, visit the help center.If this question can be reworded to fit the rules in the help center, please edit the question.

Let me answer this question under the assumption that we are talking about isomorphism' of groups. So I have a group $G$ containing two isomorphic subgroups $H$ and $K$. When can I extend the isomorphism between $H$ and $K$ so that it becomes an automorphism of $G$? The answer is that usually you can't.
In particular the model theoretic concept of homogeneity is relevant here: roughly speaking (because I am no model theorist) a class of structures is homogeneous if whenever you have three structures $H$, $K$ and $G$ in the class such that (a) $H$ and $K$ are substructures of $G$ and (b) $H$ is isomorphic to $K$, then an isomorphism between them can be extended to an automorphism of $G$.
An example of a class of groups that is homogeneous would be $\mathcal{E}_p$, the class of elementary abelian $p$-groups. That the isomorphism is always extendible follows from the fact that any linearly independent set of vectors in a vector space can be extended to a basis.
To return to the original question: in addition to Bruno's example given above, here is another: $G=C_2\times C_4$ and $H$ and $K$ are cyclic subgroups of order $2$. Take $H$ to lie in a subgroup $C_4$ and $K$ to not lie in such a subgroup. Note that $G$ is abelian so $H$ and $K$ are normal, which deals with your second question.