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Is there a general method to prove that an infinite subgroup of a group has finite index? Or, in other words, to prove that the quotient group is finite? I am particularly interested in classical groups, such as GL(n), SL(n), etc, over a nonarchimedean local field but I am looking for a general method, if there exists one.

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6 Answers 6

This is a somewhat tautological answer, but: if you can show that the subgroup contains the kernel of a finite representation (i.e. a homomorphism to a finite group), you're done. Intuitively: "I only need a finite number of things to go my way in order to belong to this subgroup."

If the group (or some representation of that group) is compact in some topology, and the subgroup contains the connected component of the identity (or an open neighbourhood of the identity), you're also done.

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For this type of problem, "general methods" tend to be quite general indeed, but here are two ideas:

1) Find an action of $G$ on a finite set $X$ and an element $x \in X$ such that $H$ is the stabilizer of $x$. Then, by the Orbit-Stabilizer Theorem, $G/H$ is isomorphic to the orbit space $Gx$, so is finite.

2) Find a finite group $X$ and a homomorphism $f: G \rightarrow X$ such that $H$ contains the kernel of $f$. Then $f: G/\operatorname{ker}(f) \hookrightarrow X$, so $\operatorname{ker}(f)$ has finite index, so $H$, which contains $\operatorname{ker}(f)$, has finite index.

Note that both of these will, in principle, always work. In Case 1, take $X = G/H$. In Case 2, let $H' = \bigcap_{g \in G} gHg^{-1}$ be the normal core of $H$. It is easy to show that (since $H$ has finite index), $H'$ is a finite index normal subgroup of $G$ which is contained in $H$. Take $X = G/H'$ and $f$ to be the quotient map.

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I'm not sure if you're interested in classical groups over Q_p or over Z_p; in the latter case, one can often show that a closed subgroup H of G is in fact the whole group G once you know it projects surjectively onto some explicit finite quotient of G. See, for instance, [IV.3.4, Lemma 3] of Serre's "Abelian l-adic representations and elliptic curves." I explain how this works for GL_n (this is definitely not original to me, I just wanted the paper to be self-contained) in Lemma 3 of this paper about K3 surfaces.

Of course this won't work unless you know for some reason that H is closed; for instance, two random elements of GL_2(Z_p) will (I think) almost always generate a (discrete) free subgroup of GL_2(Z_p) which is dense, i.e. which projects surjectively onto every finite quotient, but which is obviously not finite-index.

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This may be tangentially related: a subgroup H of a nilpotent group G is in fact the whole group if it projects surjectively onto the abelianisation of G. –  Terry Tao Oct 26 '09 at 16:49
    
Good point; similarly, if H and G are finitely generated pro-p groups, all you need is for H to surject onto the finite quotient G^{ab} / p G^{ab}. –  JSE Oct 26 '09 at 17:05

From a computational perspective, you could try coset enumeration (or Knuth-Bendix) over the candidate subgroup (starting with generators given in terms of a known presentation of the overgroup). If it terminates, then the subgroup has finite index.

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I often try to show that a subgroup contains another subgroup known to be of finite index. There are various refinements. One technique is to prove that your subgroup contains an intersection of a finite number of subgroups, each of which is known to have finite index. For example, suppose you know that the central quotient G/Z(G) of a group G is finite. (One way to show this is to show that G is a union of finitely many abelian subgroups.) If it is too hard to show that your subgroup H contains the centre, but you can show that H contains the intersection of Z(G) with the derived subgroup [G,G] of G, then the index of H in G is finite.

Another idea is to try to show that (e.g.) every nilpotent quotient of your group is finite, and then show that your subgroup must contain some term of the lower central series. (You could replace nilpotent with soluble and the LCS with the derived series, and so on.)

This may not count as a general method; perhaps it is more of a "trick", but I've seen it used to good effect in proving some commutativity theorems for groups and rings. If you can show that your subgroup is a union of (at most) two subgroups known to have finite index, then you are done.

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You could try computing the virtual cohomological dimension of the ambient group, and then the subgroup. If they're torsion free, you only need to know the cohomological dimensions, as a theorem of Serre tells you that a torsion free group and its finite index subgroups have the same dimension. (You'd be surprised how often this works.)

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I guess that's more of a "show it has infinite index" test, but still useful. –  Richard Kent Nov 4 '09 at 0:47
    
On the other hand, in a lot of settings, infinite index subgroups will have cd smaller than the ambient group. –  Richard Kent Nov 4 '09 at 0:53

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