Downward induction on upper central series

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This page is a survey article about a proof method or constellation of methods commonly used in group theory. The method is sufficiently vague that it cannot be classed as a "fact" or abstracted into a "theorem" but sufficiently precise that we can still make reasonably unambiguous identifications of proofs that use the method. It builds on the paradigm of the principle of mathematical induction.
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Downward induction on upper central series is an approach used to prove statements about nilpotent groups using the upper central series. The idea is that we start with a nilpotent group G of class c, then construct a statement with a parameter n involving the subgroup Z^n(G) such that the following hold:

  • The case n = c, i.e., the case where the subgroup is Z^c(G) = G, is readily established.
  • If we establish the statement for all n with 0 \le n \le c, the final result we seek is established (in some cases, we only need to establish the statement for 1 \le n \le c).
  • We can establish a "downward" inductive step that proves the statement for n = i - 1 assuming the statement for n= i.


Related methods

Approach used for the inductive step

Piecing together extensions: a parallelizable method

Further information: Inductive proof methods for the ascending series corresponding to a subgroup-defining function

This is a method in that it does not use the dynamics of the upper central series per se. The approach involves:

  1. Proving that the statement is true for the center in any group (of the appropriate type).
  2. Proving that the statement is preserved on taking extensions at the bottom end. This may, for instance, correspond to establishing that a particular property is a quotient-transitive subgroup property (or something somewhat more restrictive than that, such as being quotient-transitive when the smaller subgroup is central) or establishing that a group property is preserved on taking central extensions.

Step (1) -- the proof for the center -- may use a lot of facts about the precise definition of the center, and hence make the result hard to establish for other series. However, it is parallelizable in the sense that the result for each quotient Z^{i+1}(G)/Z^i(G) inside G/Z^i(G) is established independently, and then the findings are pieced together using quotient-transitivity. This could also be adapted to give a proof using upward induction on upper central series when the group is a nilpotent group.

The situation where we need to use downward induction and cannot use upward induction is the situation where we can only establish the result for central extensions rather than arbitrary extensions. If we can establish that a result is true for the center in any group, and it is true on taking central extensions, and we want to reach the whole group, we can extend from Z^c(G)/Z^{c-1}(G) to Z^c(G)/Z^{c-2}(G) to Z^c(G)/Z^{c-3}(G), because each time, we are adding a new central subgroup at the bottom. We cannot proceed upwards because when adding a new thing on top, the extension in question is not in general central.

We may be able to rescue upward induction, however, if we treat the group itself as a variable parameter in the statement being proved by induction rather than viewing the group as fixed at the outset. In other words, we could use upward induction instead, but we'd need to make the statement "for all (appropriate) groups" rather than restrict locally to the context of the particular group G.

The multihomomorphism approach

The inductive step typically involves looking at the left-normed iterated commutator as a multihomomorphism of the form:

Z^i(G) \times G \times G \times \dots \times G \to G

where the number of occurrences of G is i - 1. We note that the kernel of this for the first coordinate is precisely Z^{i-1}(G). This fact, along with multilinearity, is typically used to execute the proof.

We describe the key method used in the inductive step:

Example proof How we use the generalization
upper central series members are completely divisibility-closed in nilpotent group We use the multilinearity to "shift" the p^{th} root from one input in the iterated commutator multihomomorphism to another. Note that with a little trick, we can get away with issues of non-uniqueness of divisibility.
upper central series member operator commutes with root set operator for torsion-free nilpotent group We use the multilinearity to argue that if we take p^{th} root on the first coordinate in an input tuple that goes to the identity element, then the output from the new input tuple is a p^{th} root of the identity element. We now use the torsion-freeness to show that in fact the output is still the identity element.

Necessity of using the upper central series and breakdown for the lower central series

If the statement that we are trying to prove is itself an assertion regarding the upper central series, then obviously we need to use the upper central series for the proof. The more relevant question would be: "what aspect of the proof specifically uses the fact that we are dealing with the upper central series, and would break down for the lower central series?"

The answer to this is that the upper central series member is the full kernel of the commutator multihomomorphism. Explicitly, recall the left-normed iterated commutator as a multihomomorphism of the form:

Z^i(G) \times G \times G \times \dots \times G \to G

where the number of occurrences of G is i - 1.

We note that the set of elements in Z^i(G) for which the output is always trivial (regardless of the other inputs) is precisely Z^{i-1}(G).

The lower central series and other central series are characterized by the problem that the subgroups in these series could be too "small" and hence, when we construct new candidate elements that are in the kernel, they are guaranteed to be in the appropriate upper central series member but need not be in the appropriate lower central series member.

There may, however, be another proof method that works for the lower central series.

Possibilities for generalization

What we'd like to do Whether it's possible and how we'd do it
Generalize the statement to non-nilpotent groups (i.e., remove the assumption that the group is nilpotent) while changing as little of the rest of the statement as possible. For the "piecing together" approach, this is usually possible. We just begin by declaring what upper central series member we are aiming to prove the statement for, then induct downward starting from there. Alternatively, we just rethink of it as upward induction with the group itself as a parameter rather than fixed in the backdrop.
For the multihomomorphism approach, this is usually not possible, because downward induction crucially relies on the upper central series actually reaching the group. This is in contrast to upward induction on upper central series, where (typically) some of the implications work in the non-nilpotent case, because it is possible to kickstart the induction.
Generalize the statement to a situation involving a group and a subgroup For this, we typically assume that the subgroup is a normal subgroup contained in the hypercenter, even if the group is not nilpotent. Depending on how we generalize, though, we may need that the group is nilpotent.