Upward 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.
View other survey articles about proof methods|View survey articles about proof methods that build on the principle of mathematical induction

Upward induction on upper central series is an approach used to prove statements about groups using their upper central series. The method is typically used for nilpotent groups.

We construct a statement with a parameter n involving the upper central series member Z^n(G) such that the following hold:

  • The case n = 0 is readily established. In some cases, we may start with n = 1.
  • We can establish an "upward" inductive step from n = i to i + 1 that proves the statement for n = i + 1 assuming its truth for n = i.
  • If we are interested in going to the transfinite upper central series, we also need a procedure for handling the limit ordinals. This may involve proving something about joins or unions.
  • In the case where G is nilpotent of class c, reaching n = c may carry some additional significance in terms of establishing a result for the whole group.

Examples

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

For an example of this method in action, see upper central series members are quotient-powering-invariant

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).

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 downward induction on upper central series when the group is a nilpotent group. In fact, there are situations involving restriction to central extensions where downward induction may be the only choice. 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.

The bihomomorphism approach

For an example of this, see equivalence of definitions of nilpotent group that is torsion-free for a set of primes (the (3) implies (4) implication is the one that uses induction)

The inductive step typically uses the bihomomorphism obtained via the commutator map:

Z^{i+1}(G)/Z^{i-1}(G) \times G/Z^{i-1}(G)\to Z^i(G)/Z^{i-1}(G)

The kernel for the first coordinate is Z^i(G)/Z^{i-1}(G) for obvious reasons. Note that this precise characterization of the kernel uses the definition of the upper central series and does not generalize to other central series, where the kernel may be a lot larger than the intermediate central series member.

We then combine this with information we have already gathered in the inductive hypothesis, which typically pertains to Z^i(G)/Z^{i-1}(G) (and smaller quotients), to deduce something about Z^{i+1}(G)/Z^i(G).

Such an approach is not so easy to parallelize because we are crucially using the inductive hypothesis in the inductive step, rather than getting separate information at different levels that we simply piece together.

Proving statements for the transfinite upper central series

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

For the "piecing together" type proofs, the necessity of using the upper central series arises from the necessity of using the center.

For the bihomomorphism approach proofs, the key fact is that the upper central series member gives the full kernel of the bihomomorphism.