Verifying conjectures for finite groups

This article aims to look at the general question: given a conjecture stating that something is true for all finite groups, how do we go about verifying, proving, or disproving the conjecture?

Existential versus universal type
A conjecture related to finite groups typically has either of these formats:


 * Existential type: Start with a finite group. Then, there exists a structure such that ...
 * Universal type: Start with a finite group. Then, for all structures of some type, ...

Conjectures of existential type are, for instance, the conjecture for the inverse Galois problem, which states that for any group, there exists a field extension of $$\mathbb{Q}$$ having that as the Galois group. More generally any conjecture stating that every group is realizable in some fashion, is of the existential type.

Proving conjectures of existential type may involve making an explicit construction of a structure satisfying the conditions posited.

Conjectures of universal type are, for instance, the finite-extensible automorphisms conjecture, which states that every extensible automorphism of the group is inner.

Conjectures of universal type can often be converted to conjectures of existential type, often through a clever manipulation of the ideas.

Proving results about the underlying property
When confronted with a conjecture that one claims is true for all finite groups, there is a natural question of where to start. Typically, though, there is a way to start. One simple approach to start is to try to prove results about the property of being a finite group that satisfies the conjecture. Ideally, we want to prove that every finite group satisfies the conjecture. So if we can show that this property is sufficiently nice, we can reduce the proof to proving for some particular finite groups.

For instance, if we can show that the property is preserved upon taking extensions of groups, it suffices to concentrate on simple groups. If we show that the property is preserved on taking direct products (viz, is finite direct product-closed), it suffices to concentrate on directly indecomposable groups. In practice, most of the properties associated with conjectures cannot easily be shown to have any of these metaproperties.

Proving results for some basic cases
Even if we do not establish any of the nice metaproperties, it may still help to prove that all sufficiently well-understood and easy-to-manipulate groups have the given property. A quick checklist of starting points:


 * Cyclic groups and Abelian groups are particularly easy to manipulate, because for these groups, we completely understand the structure (thanks to the structure theorem for Abelian groups). We can also conveniently describe the representation theory, and other aspects, for these groups.
 * A group of prime power order may also be relatively easy to manipulate, and hence, it may be possible to handle all nilpotent groups. However, the general structure theory and representation theory of nilpotent groups is not well-understood, so nilpotent groups offer an advantage only in specific situations where we can exploit what we know about nilpotent groups.
 * The finite simple groups are another good starting point. There are two good things about these: the first is that all homomorphisms from them are injective, and hence all their representations, for instance, are faithful. Secondly, although their representation theory and other structural aspects are by no means simple, they have been worked out in full detail in work leading up to and following the classification.
 * After the finite simple groups, the next ones may be the finite characteristically simple groups, which are direct powers of finite simple groups.
 * In a different direction, solvable groups may be easy to handle. Again, not much is known about the general structure theory or representation theory of solvable groups, so any approach to handling them must somehow exploit whatever little is known.
 * Another way of controlling what groups to start with is by controlling the number of prime divisors, and their type. Certain conjectures may be easier to settle for odd-order groups, while some may be easy for groups with very few prime divisors. Some conjectures may be particularly easy for groups where the exponent of each prime divisor is small.
 * Some conjectures may be easier to settle for groups having Sylow subgroups or Hall subgroups with certain special properties.

Constructive approach
Often, proving a conjecture of existential type involves constructing an explicit algebraic object. In questions of realizability, this explicit algebraic object is directly given, while in other cases, it is more subtle. Here are some tools for explicit construction:


 * If the group is realized as sitting inside or mapped to some matrix group over a field (through a linear representation) or more generally is mapped to the automorphism group of a module, we may get our algebraic object from that mapping.

The construction or proof of its working often relies on the many special properties of matrix groups over fields. Sometimes, we may need to consider the collection of all linear representations and use the powerful orthogonality theorems and conjugacy class-representation duality.


 * If the group is realized as acting on a combinatorial object, we may be able to use the combinatorics and the theory of permutation representations to effect the required construction.