Induction for subgroups of finite groups

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This article discusses general methods for proving results that involve finite groups and subgroups. These results may be of the form: Given any finite group G and any subgroup H, a certain fact is true about G and H.

Alternatively, it may be a statement of the form: Given any finite group G, and two subgroups H, K of G satisfying certain conditions, a certain fact is true about G,H,K.

Other kinds of statements make an assertion about all subgroups of a particular order or index or of a particular type in a group.

Avoid proofs by induction

Indication that a proof by induction may be needed

Proofs by induction should generally be used only as a last resort, after all direct reasoning has failed. Some indicators that a proof by induction might be needed:

  • The statement itself uses the language of finiteness, by talking about the orders and indices, or about divisors of these numbers.
  • There is no intrinsic or direct proof.
  • The result fails for infinite groups, and does not seem to have an obvious analogue for infinite groups.
  • The result requires one or more constructive choices.

Before starting on the induction

Before starting a proof by induction, it is good to attempt a number of easy cases through direct proofs. This helps in two ways:

  • In some situations, it turns out that a proof by induction is unnecessary.
  • In other situations, the easy cases suggest an appropriate parameter of induction.

After succeeding in the proof

After completing the proof by induction, it is good to review the proof to see if induction can be eliminated from it. In some cases, this may require proving other results that are not naturally suggested initially. Those results may in turn use induction, but in a less noticeable way.

Possibilities for the parameter of induction

We know, by Lagrange's theorem, that if H is a subgroup of G, then:

|G| = |H|[G:H].

Thus, there are three related numbers describing the group and the subgroup: the order of the whole group G, the order of the subgroup H, and the index [G:H].

This leads to different possible ways to induct.

Induct on the order of the group

Here, we induct on the order of the group. In other words, we assume that for all groups of order strictly less than the order of G, the statement is true, and we try to prove it for G. We do not make any assumption about the result being true for other orders of H for that given order of G.

Induct on the order of the subgroup

Here, we induct on the order of H. In other words, we assume that for all groups of order strictly less than the order of H, the result is true whenever that group plays the role of the subgroup. We do not make any assumptions about the order of G.

Induct on the index of the subgroup

Here, we induct on the index [G:H].

Double inductions

With these three main parameters of induction, we can combine them in different ways:

  • Induct first on the order of the group, and second, on the index of the subgroup.
  • Induct first on the order of the group, and second, on the order of the subgroup.

Other combinations are possible too.

Other order-like quantities and index-like quantities

When dealing with a subgroup, we may want to induct based on other numerical invariants associated with the subgroup. Some of these numerical invariants, such as the exponent of the subgroup, may be order-like in nature: bigger groups have bigger exponent, and an induction on the exponent relies on proving the result for smaller subgroups before proving it for larger ones.

Other quantities may be index-like: they may be smaller for larger subgroups. Inducting on such quantities means that we first prove the result for larger subgroups and then use that to prove the result for smaller subgroups.

Quantities that are neither order-like nor index-like

There are some quantities associated with subgroup that do not have a direct or clear relation with subgroup containment. When a proof relies on induction on such a quantity, the proof may be harder to come by. Here are some examples:

  • The subnormal depth of the subgroup: This quantity is 1 for every normal subgroup, so it is small both for the trivial subgroup and for the whole group. Subnormal depth has no direct relation with containment, so a proof by induction on subnormal depth cannot be translated to a proof by induction either on order or on index.
  • The order or index of the normalizer of the subgroup: The normalizer of a subgroup is not monotone. A smaller subgroup may have a smaller normalizer or a bigger normalizer. Proofs that rely on induction on the order or index of the normalizer cannot be translated to induction on the order or index of the subgroup.
  • A measure of deviation from some property, like normality: For instance, the index of the subgroup in its normal closure, or the index of the normal core in the subgroup, or the index of the normal core in the normal closure.

Determining the parameter of induction

Making a choice between different parameters of induction can be hard. Here are some heuristics:

  • Before beginning the proof by induction, make some preliminary observations that settle the result in easy cases, or indicate equivalent formulations of the problem.
  • See what the easy cases are and what parameter of induction would render them as the base cases to build upon.

Here are some examples.

Subgroups of prime power order

Further information: Congruence condition on number of subgroups of given prime power order