Thompson's critical subgroup theorem

History
The critical subgroup theorem was first proved in the joint paper on the odd-order theorem by Walter Feit and John Griggs Thompson. The part of the paper containing this theorem (Chapter 2, Lemma 8.2, see also the references below) is generally attributed to Thompson.

The term critical subgroup appeared in Gorenstein's book on Finite Groups (see also the references below).

General statement
Let $$G$$ be a group of prime power order, i.e., a finite $$p$$-group for some prime $$p$$. Then, $$G$$ has a critical subgroup, i.e., a characteristic subgroup $$H$$ satisfying the following four conditions:


 * 1) $$\Phi(H) \le Z(H)$$, viz., the Frattini subgroup is contained inside the center (i.e., $$H$$ is a Frattini-in-center group ).
 * 2) $$[G,H] \le Z(H)$$ (i.e., $$H$$ is a commutator-in-center subgroup  of $$G$$).
 * 3) $$C_G(H)= Z(H)$$ (i.e., $$H$$ is a self-centralizing subgroup  of $$G$$).
 * 4) $$H$$ is coprime automorphism-faithful  in $$G$$: If $$\sigma$$ is a non-identity automorphism of $$G$$ such that the order of $$\sigma$$ is relatively prime to $$p$$, then the restriction of $$\sigma$$ to $$H$$ is a non-identity automorphism of $$H$$.

Note that since characteristic and self-centralizing implies coprime automorphism-faithful, a characteristic subgroup satisfying condition (3) automatically satisfies condition (4). Thus, it suffices to show conditions (1)-(3).

Analogues for other kinds of groups

 * Analogue of critical subgroup theorem for finite solvable groups
 * Analogue of critical subgroup theorem for infinite abelian-by-nilpotent p-groups

Applications

 * Odd-order p-group has coprime automorphism-faithful characteristic class two subgroup of prime exponent
 * Classification of finite p-groups of characteristic rank one

Other related facts

 * Abelian Frattini subgroup implies centralizer is critical

Analysis
While Thompson's critical subgroup theorem is constructive, it does not necessarily yield all possible critical subgroups. In fact, it yields critical subgroups satisfying two additional constraints: the center is maximal among abelian characteristic subgroups and, moreover, a critical subgroup obtained through this procedure is completely determined by its center, while there may be other critical subgroups with the same center. A critical subgroup that can arise through the constructive procedure of this theorem is termed a constructibly critical subgroup. It turns out that every abelian critical subgroup is constructibly critical.

The fact that there is no unique choice of critical subgroup makes critical subgroups different from other characteristic subgroups we typically encounter. More information is available at analysis of Thompson's critical subgroup theorem.

General facts

 * 1) uses::Maximal among abelian normal implies self-centralizing in nilpotent
 * 2) uses::Characteristic and self-centralizing implies coprime automorphism-faithful

Facts about characteristicity
The fact numbers given here are for reference in the proof, and have no deeper significance.

Facts about Omega-1

 * 1) uses::Omega-1 of center is normality-large: For a finite $$p$$-group $$P$$, the intersection of $$\Omega_1(Z(P))$$ with any nontrivial normal subgroup of $$P$$ is a nontrivial subgroup (in fact, it is a nontrivial normal subgroup).

Proof
Given: A finite $$p$$-group $$G$$

To prove: $$G$$ has a critical subgroup.

Note that by fact (2) in the list of general facts (characteristic and self-centralizing implies coprime automorphism-faithful) it suffices to find a characteristic subgroup satisfying conditions (1)-(3).

If there exists a characteristic subgroup maximal among abelian normal subgroups
We first consider the case that there exists a subgroup $$M$$ of $$G$$ maximal among Abelian normal subgroups, such that $$M$$ is also a characteristic subgroup. By fact (1) in the list of general facts, we see that $$M$$ is a self-centralizing subgroup. We check all the conditions for $$M$$:


 * 1) Frattini-in-center group condition: $$\Phi(M) \le Z(M)$$: This condition is satisfied trivially because $$M$$ is Abelian.
 * 2) Commutator-in-center subgroup conidtion: $$[G,M] \le Z(M)$$: By normality of $$M$$, $$[G,M] \le M$$, and by Abelianness of $$M$$, $$M = Z(M)$$. Thus, $$[G,M] \le Z(M)$$.
 * 3) Self-centralizing subgroup condition: $$C_G(M) \le M$$: This condition is satisfied by assumption.

In the other case: setting up the subgroups


We now begin the proof.

Proving the conditions for criticality
Our goal is now to show that $$C$$ satisfies the conditions for being a critical subgroup. Recall that we have established that $$C$$ is a characteristic subgroup of $$G$$ and that $$Z(C) = K$$. Also, recall that:

$$H = C_G(K), \overline{H} = H/K, \overline{L} = \Omega_1(Z(\overline{G}), C = H \cap L, \overline{C} = C/K$$.

We check the conditions one by one:


 * 1) Frattini-in-center group condition: We need to show that $$\Phi(C) \le K$$: For this, observe that $$\overline{C}$$ is contained in $$\Omega_1(Z(\overline{G}))$$, which is elementary Abelian (it is contained in an Abelian group $$Z(G)$$, and is generated by elements of order $$p$$). Thus, $$C/K$$ is elementary Abelian, and we get $$\Phi(C) \le K$$.
 * 2) Commutator-in-center subgroup condition: We need to show that $$[G,C] \le K$$: Modulo $$K$$, the image of $$C$$ is contained in $$\Omega_1(Z(\overline{G}))$$, which is contained in $$Z(\overline{G})$$. Hence $$[\overline{G}, \overline{C}] \le [\overline{G},Z(\overline{G})]$$, which is trivial. Hence $$[G,C] \le K$$.
 * 3) Self-centralizing subgroup condition: We need to show that $$C_G(C) \le C$$: We do this by contradiction. Since the proof is somewhat long, we do it in the table below.

ASSUMPTION: Suppose $$Q = C_G(C)$$, such that $$Q$$ is not in $$C$$.

Journal references

 * : Thompson's critical subgroup theorem appears as Lemma 8.2, Section 8 (Miscellaneous preliminary lemmas) in Chapter 2, Page 795.