Arithmetic and normal subgroup structure

This article discusses the relation between the arithmetic structure (the structure of subgroups whose orders have various prime divisors) and the normal structure (the structure of normal subgroups, characteristic subgroups, and subnormal subgroups) in finite groups.

The starting point of a systematic exploration of the relationship was the paper Arithmetic and Normal Subgroup Structure by Wielandt and Huppert, that appeared in ''Proc. Symposia in Pure Math'', published by the American Mathematical Society in 1962.

Starting points
A purely arithmetic property of a subgroup is a property that depends only on the order of the group, and the order of the subgroup. The starting point of arithmetic properties of subgroups is the following key result, known as Lagrange's theorem:

''The order of a subgroup divides the order of the group (for a finite group). The ratio is the index of the subgroup''

It is not true that for every number dividing the order of the group, there is a subgroup with that number as order, but in certain caes, we can guarantee the existence of subgroups of certain orders.

One question is: if we have two subgroups of the same order, are they similar in a stronger, group-theoretic sense?

Another question is this: if the order of one subgroup divides the order of the other, is there some way in which the first subgroup is smaller than, or contained in, the other subgroup? For a cyclic group, this is true, but it is not true in general.

Notions of resemblance
We can consider the following progressively stronger notions of resemblance for subgroups in a group:


 * Having the same order (and hence, also having the same index)
 * Being isomorphic as groups. In other words, the two subgroups are abstractly isomorphic
 * Being automorphs: There exists an automorphism of the group mapping one subgroup isomorphically onto another
 * Being conjugate subgroups: There exists an inner automorphism of the group mapping one subgroup isomorphically onto another
 * Being equal: The two subgroups actually coincide

We have the following related subgroup properties:


 * Order-unique subgroup: There's only one subgroup of that order
 * Order-conjugate subgroup: Any other subgroup of that order is conjugate to it
 * Order-automorphic subgroup: Any other subgroup of that order is an automorph
 * Order-isomorphic subgroup: Any other subgroup of that order is isomorphic
 * Isomorph-free subgroup: There's no other isomorphic subgroup
 * Isomorph-conjugate subgroup: Every isomorphic subgroup is conjugate
 * Isomorph-automorphic subgroup: Every isomorphic subgroup is an automorph
 * Characteristic subgroup: There's no other automorph
 * Automorph-conjugate subgroup: Every automorph is a conjugate
 * Normal subgroup: There's no other conjugate subgroup

Part of the job of relating the arithmetic and normal subgroup structure is investigating the conditions on the order of the group and subgroup, as well as the properties that the group must satisfy, for the subgroup to satisfy various versions of the properties listed above.

Notions of domination
Domination is stronger than resemblance: here, we are given two subgroups $$H_1$$ and $$H_2$$ of a group $$G$$. We would like to know whether one is smaller than the other. The following progressively stronger notions exist:


 * The order of the first subgroup, divides the order of the second subgroup ($$|H_1| \mid |H_2|$$)
 * The first subgroup is isomorphic to a subgroup of the second subgroup ($$H_1 \cong H_3 \le H_2$$)
 * There is an automorphism of the whole group mapping the first subgroup isomorphically to a subgroup of the second ($$\exists \sigma \ \in Aut(G), \ \sigma(H_1) \le H_2$$)
 * There is an inner automorphism of the whole group mapping the first subgroup isomorphically to a subgroup of the second ($$\exists \ g, gH_1g^{-1} \le H_2$$)
 * The first subgroup is contained in the second ($$H_1 \le H_2$$)

We in general want to study the question of what constraints exist on the orders of $$H_2$$ and $$G$$ (and the nature of $$G$$) such that the first of the above notions implies some of the lower ones. The particular notion of importance is that of an order-dominating subgroup: a subgroup for which the first condition implies the fourth i.e. any subgroup whose order divides it, is conjugate in the whole group to a subgroup of it.

Sylow's theorem
Sylow's theorem is one of many ECD results, namely, it establishes that a certain kind of subgroup exists, any two such subgroups are conjugate, and every smaller subgroup is contained in one of those conjugates (domination).

To state Sylow's theorem, we first define the notion of a $$p$$-group. A $$p$$-group (called a group of prime power order) is a finite group whose order is a power of the prime $$p$$. The trivial group is considered a $$p$$-group for every $$p$$.

A $$p$$-Sylow subgroup is a $$p$$-subgroup in a finite group whose index is relatively prime to $$p$$. In other words, its order is the largest power of $$p$$ dividing the order of the group. Sylow's theorem states the following:


 * Existence: If $$G$$ is a finite group, and $$p$$ is any prime, then $$p$$-Sylow subgroups exist (if $$p$$ does not divide the order of $$G$$, then the $$p$$-Sylow subgroup is trivial).
 * Conjugacy: Any two $$p$$-Sylow subgroups in $$G$$ are conjugate.
 * Domination: Any $$p$$-subgroup of $$G$$ is contained in a $$p$$-Sylow subgroup.

The second and third part can be combined into the statement that an order-dominating $$p$$-Sylow subgroup exists. There are more parts to Sylow's theorem, which place number-theoretic restrictions on the number of $$p$$-Sylow subgroups (it should be 1 mod $$p$$, and should divide the index).

Sylow's theorem thus gives a partial converse to Lagrange's theorem (existence of subgroups of certain orders). It also shows that Sylow subgroups are order-dominating, hence order-isomorphic, order-automorphic, order-conjugate, isomorph-automorphic, isomorph-conjugate, and automorph-conjugate. Further corollaries include the fact that Sylow subgroups are pronormal (this follows from being isomorph-conjugate).

More existence results
The following result is true in $$p$$-groups: there exist subgroups of every order dividing the order of the group. This, along with Sylow's theorem, tells us that for a finite group $$G$$, and given any prime power that divides the order of $$G$$, there exists a subgroup with that prime power as its order.

Tying up with normal structure
The conjugacy and domination parts of Sylow's theorem already hint at the fact that the normal structure of the group could be closely related to its arithmetic structure. For instance, the following facts follow readily from Sylow's theorem:


 * If a Sylow subgroup is subnormal, then it is normal. This follows from the fact that Sylow subgroups are pronormal, and subnormal and pronormal implies normal. In fact, if a Sylow subgroup is normal, it is also characteristic, even fully characteristic.
 * A finite nilpotent group is a direct product of its Sylow subgroups, and in general, a finite group is nilpotent iff every Sylow subgroup is a direct factor, iff every Sylow subgroup is normal. Thus, the theory of finite nilpotent groups breaks down completely to the study of finite $$p$$-groups.
 * For a nilpotent group, there exist subgroups of all orders (the converse is not true; there are non-nilpotent groups with subgroups of all orders).

Note that the latter result does not state that if a particular Sylow subgroup is normal, then it is a direct factor. As we shall see, this is really the subject of Hall theory.

Hall theory
The idea behind Hall theory is to split off a group into pieces that do not affect each other. A group can be split as a direct product, and more generally as a semidirect product. The hope is that if we have a subgroup whose order and index are relatively prime, we can split the group off as a semidirect product involving that.

A Hall subgroup in a finite group is defined as a subgroup whose order and index are relatively prime. The property of being Hall is more robust than the proprty of being Sylow, for instance, it is an identity-true subgroup property (every group is Hall in itself).

In solvable groups
The first result states for for a finite solvable group the ECD condition holds for $$\pi$$-Hall subgroups. For a set of primes $$\pi$$, a Hall $$\pi$$-subgroup is a subgroup such that the set of prime divisors of its order is contained in $$\pi$$, and the set of prime divisors of its index is disjoint from $$\pi$$. If $$\pi$$ is a subset of the set of prime divisors of the whole group, this is the same as saying that the prime divisors of the order of the subgroup are precisely the elements of $$\pi$$.

More explicitly, the result is:


 * Existence: For every prime set $$\pi$$, $$\pi$$-Hall subgroups exist
 * Conjugacy: Any two $$\pi$$-Hall subgroups are conjugate
 * Domination: Any $$\pi$$-subgroup (subgroup whose set of prime divisors is contained in $$\pi$$, is contained in a $$\pi$$-Hall subgroup

Thus $$\pi$$-Hall subgroups are order-dominating, so all the corollaries hold (for instance, they are pronormal, automorph-conjugate).

However, for finite solvable groups, it is no longer true that subgroups of every order dividing the order of the group exist. Interestingly, as we shall see a little later, the converse is true i.e. if subgroups of every order exist, the subgroup is solvable.

Schur-Zassenhaus theorem
The Schur-Zassenhaus theorem gives a somewhat different situation where we can guarantee the existence of a Hall subgroup:


 * Existence: If $$H$$ is a normal Hall subgroup of a group $$G$$, $$H$$ has a permutable complement $$K$$.
 * Conjugacy: Any two complements to $$H$$ are conjugate in $$G$$

Note that it is clear that any complement to $$H$$ is a Hall subgroup on the complementary set of prime divisors, and it is also clear that any two complements are isomorphic (because any complement is isomorphic to the quotient group $$G/H$$). The hard part is showing the existence of a complement, and showing that the complements are actually conjugate and not just isomorphic. What the Schur-Zassenhaus says is that if the complementary set of prime divisors exhibits a normal Hall subgroup, then the original set of prime divisors admits a Hall subgroup and a unique conjugacy class of Hall subgroups (so the Hall subgroups are order-conjugate).

Hall's theorem
This result, often called Philip Hall's theorem, states that a finite group is solvable iff there exist Hall subgroups of all possible orders (this does not guarantee the existence of subgroups of all possible orders dividing the order of the group). Thus, Hall's theorem establishes a converse to the ECD conditions for Hall subgroups in solvable groups.

General theory for Hall subgroups
In general Hall subgroups need not exist, need not form a single conjugacy class, and not every $$\pi$$-subgroup need be contained in a Hall $$\pi$$-subgroup. In fact, almost all the corollaries we saw for Sylow subgroups break down for Hall subgroups, such as:


 * Hall not implies order-isomorphic
 * Hall not implies isomorph-automorphic
 * Hall not implies automorph-conjugate

Relating arithmetic structure and normal structure
It is the conjugacy and domination parts of results from Sylow theory and Hall theory that help us get a finer hold on the relation between the arithmetic structure and the normal structure on finite groups. One of the key uses of arithmetic structure is the concept of $$p$$-local structure: build the group using information about $$p$$-local subgroups: subgroups that are obtained as normalizers of $$p$$-groups. Of particular interest are the normalizers of the Sylow subgroups, and the relation of those with the whole group.

Since a Sylow subgroup is pronormal, its normalizer is abnormal, and in particular self-normalizing.