Nilpotent versus solvable

This survey article compares, and contrasts, the following group properties: nilpotent group versus solvable group
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Introduction

A first glance at the definitions of nilpotent group and solvable group may seem to suggest that the definitions are extremely similar, and so the group properties of nilpotence and solvability are not very different. Although the properties behave similarly in some respects, the flavor of nilpotent groups is very different from that of solvable groups. This article attempts to provide some insight into the differences between the two group properties.

Definitions

Nilpotent group

Further information: nilpotent group, finite nilpotent group

A group is termed nilpotent if it satisfies the following equivalent conditions:

• Its upper central series terminates after finitely many steps at the whole group
• Its lower central series terminates after finitely many steps at the trivial subgroup

For finite groups, there are other equivalent formulations:

• Every Sylow subgroup is normal
• The group is a direct product of its Sylow subgroups
• Every maximal subgroup is normal
• There is no proper self-normalizing subgroup
• Every subgroup is subnormal

As we shall see in the discussions below, nilpotence is more of a global constraint. It imposes strong restrictions on the way any subgroup can be embedded in the group (namely, it has to be subnormal).

Solvable group

Further information: solvable group, finite solvable group

A group is termed solvable if it satisfies the following equivalent conditions:

• Its derived series terminates after finitely many steps at the trivial subgroup
• It has a normal series where all the successive quotients are Abelian
• It has a subnormal series where all the successive quotients are Abelian

As we shall see in the ensuing discussion, the constraints placed by solvability are significantly more local in nature.

Extensions and induction

Solvability: closed under extensions

Further information: induction for finite solvable groups

Solvability is closed under taking finite extensions: an extension of a solvable group by a solvable group is solvable. In fact, if we take the closure of nilpotence under finite extensions, we get solvability.

This is a powerful description of solvability and also constitutes the usual method for showing and using solvability. For instance, suppose we need to prove that any group with a particular property is solvable. Then, by the closure under taking extensions, all we need to do is find a solvable normal subgroup such that the quotient is also solvable. The same cannot be done for nilpotent groups. Finding a nilpotent normal subgroup with a nilpotent quotient group does not guarantee that the whole group is nilpotent.

Another way of saying this is that, particularly for finite groups, the property of solvability can be checked simply by checking what factors occur in the composition series. If all the factors in the composition series are simple Abelian, then the group is solvable. If even one factor is simple non-Abelian, then the group is non-solvable.

Thus, for instance, if we have a group property that is closed under taking extensions, and we prove that no simple non-Abelian group satisfies the property, then we'd have proved that any finite group satisfying the property must be solvable. For instance, the odd-order theorem shows that any odd-order group is solvable, by essentially showing that there is no simple non-Abelian group of odd order.

Nilpotence: requires more information about pasting

Further information: induction for finite nilpotent groups

To prove nilpotence, we need somewhat better information about the way the subgroups are pasted. Just knowing the composition series does not in general guarantee that the group is nilpotent.

There is one special case where we can look at the composition series to deduce the group is nilpotent: the case of a group of prime power order, or a group where all the composition factors are cyclic of the same prime order. If, however, there are multiple primes occurring in the composition series, then knowing the composition series itself does not guarantee nilpotence.

Although we cannot use induction to prove that a group is nilpotent in the same way that we can use for solvability, the reverse procedure, i.e. using nilpotence, still goes through. That is because any subgroup and any quotient of a nilpotent group is nilpotent. What fails is reasoning the other way around: just knowing that a normal subgroup is nilpotent and the quotient group is nilpotent does not guarantee that the whole group is nilpotent.

Finding small subgroups

Solvable groups: fewer guarantees

Given a finite solvable group, we can guarantee that there exists an elementary Abelian normal subgroup: take any minimal normal subgroup, and that should work.

However, solvable groups could very well be centerless, so there's in general no natural way to pick an Abelian normal subgroup. Worse, there may be no Abelian normal subgroup properly containing the center (there is still always a nontrivial Abelian characteristic subgroup, namely the penultimate term of the derived series).

Nilpotent groups: more guarantees

For nilpotent groups, we have the following facts:

• The center is always nontrivial
• In fact, the center is normality-large: its intersection with any nontrivial normal subgroup is normal
• There is always an Abelian normal subgroup properly containing the center

The last property actually falls out of another condition called being a supersolvable group: this is weaker than nilpotence for finite groups.

Arithmetic structure

Nilpotent groups: No interactions between primes

For a finite nilpotent group, the primes do not interact with each other at all. A finite nilpotent group is a direct product of its Sylow subgroups, and moreover, any subgroup of it is a direct product of its projections onto the direct factors. Thus, for all practical purposes, the theory of nilpotent groups reduces to the theory of a group of prime power order.

Solvable groups: Significant interaction between primes

In a solvable group, there is significant interaction between the primes, and this interaction is what builds the complexity of the solvable group. For instance, the symmetric group on three elements is an example of a solvable group where the 2-Sylow subgroup acts on the 3-Sylow subgroup.

For a solvable group, there is at least one prime for which the Sylow subgroup isn't normal; it could also happen that the Sylow subgroup isn't normal for any prime. In other words, we cannot always separate the study of solvable groups into different Sylow subgroups living in separate worlds and interacting through actions. It may happen that for every composition series, the primes occur in a fairly mixed-up order.