Thompson's critical subgroup theorem
This article gives the statement, and possibly proof, of a particular subgroup of kind of subgroup in a group being self-centralizing. In other words, the centralizer of the subgroup in the group is contained in the subgroup
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This article gives the statement, and possibly proof, of a particular subgroup of kind of subgroup in a finite group being coprime automorphism-faithful. In other words, any non-identity automorphism of of the whole group, of coprime order to the whole group, that restricts to the subgroup, restricts to a non-identity automorphism of the subgroup.
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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).
Statement
General statement
Let be a group of prime power order, i.e., a finite -group for some prime . Then, has a critical subgroup, i.e., a characteristic subgroup satisfying the following four conditions:
- , viz., the Frattini subgroup is contained inside the center (i.e., is a Frattini-in-center group).
- (i.e., is a commutator-in-center subgroup of ).
- (i.e., is a self-centralizing subgroup of ).
- is coprime automorphism-faithful in : If is a non-identity automorphism of such that the order of is relatively prime to , then the restriction of to is a non-identity automorphism of .
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).
Related facts
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
Analysis
Further information: Analysis of Thompson's critical subgroup theorem
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.
Facts used
General facts
- Maximal among abelian normal implies self-centralizing in nilpotent
- Characteristic and self-centralizing implies coprime automorphism-faithful
Facts about normality
Fact No. | Statement | Explanation |
---|---|---|
Norm1 | Normality satisfies image condition | The image of a normal subgroup, under a surjective homomorphism, or quotient map, is normal. |
Norm2 | Normality satisfies inverse image condition | The inverse image of a normal subgroup, under any homomorphism, is normal. |
Norm3 | Normality is centralizer-closed | The centralizer of any normal subgroup is normal. |
Facts about characteristicity
The fact numbers given here are for reference in the proof, and have no deeper significance.
Fact no. | Statement | Explanation |
---|---|---|
Char1 | Characteristic implies normal | Every characteristic subgroup is normal |
Char2 | Characteristicity is quotient-transitive | If is characteristic in , and is an intermediate subgroup such that is characteristic in , then is characteristic in |
Char3 | Characteristicity is transitive | If is a characteristic subgroup of , and is a characteristic subgroup of , then is a characteristic subgroup of |
Char4 | Characteristicity is strongly intersection-closed | An intersection of characteristic subgroups is characteristic. In particular, if are characteristic subgroups, then is also characteristic. |
Char5 | Characteristicity is centralizer-closed | If is a characteristic subgroup of , the centralizer is also a characteristic subgroup. |
Facts about Omega-1
- Omega-1 of center is normality-large: For a finite -group , the intersection of with any nontrivial normal subgroup of is a nontrivial subgroup (in fact, it is a nontrivial normal subgroup).
Proof
This proof uses a tabular format for presentation. Provide feedback on tabular proof formats in a survey (opens in new window/tab) | Learn more about tabular proof formats|View all pages on facts with proofs in tabular format
Given: A finite -group
To prove: 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 of maximal among Abelian normal subgroups, such that is also a characteristic subgroup. By fact (1) in the list of general facts, we see that is a self-centralizing subgroup. We check all the conditions for :
- Frattini-in-center group condition: : This condition is satisfied trivially because is Abelian.
- Commutator-in-center subgroup conidtion: : By normality of , , and by Abelianness of , . Thus, .
- Self-centralizing subgroup condition: : This condition is satisfied by assumption.
In the other case: setting up the subgroups
Step no. | Letter introduced | Way of choosing it | Unique choice (given previous choices)? | Relevant observations for subsequent choices |
---|---|---|---|---|
CS1 | maximal among abelian characteristic subgroups of | no | Since is characteristic, it is also, in particular, normal, so is an abelian normal subgroup of . | |
CS2 | maximal among abelian normal subgroups of containing | no | is maximal among abelian normal subgroups, and is a proper subgroup of . | |
CS3 | : the centralizer of in | yes | Since is abelian and contains , we have . In particular, . | |
CS4 | is the quotient group by | yes | This is permissible since is a characteristic subgroup, and by fact (1) on characteristic subgroups, every characteristic subgroup is normal. | |
CS5 | This is the set of elements of order dividing in the center of , and is also the socle of . | yes | This is a characteristic subgroup of | |
CS6 | The inverse image in of the | yes | contains and . We will also denote as to be consistent with the rest of the bar notation. | |
CS7 | yes | Since is contained in both and , is contained in |
We now begin the proof.
Step no. | Assertion | Facts used | Construction steps used | Proof steps used | Explanation |
---|---|---|---|---|---|
PS1 | is characteristic in | (Char2) | (CS1), (CS5), (CS6) | -- | [SHOW MORE] |
PS2 | is characteristic in | (Char5) | (CS1), (CS3) | -- | [SHOW MORE] |
PS3 | is characteristic in | (Char4) | -- | (PS1), (PS2) | [SHOW MORE] |
PS4 | is characteristic in | (Char3) | -- | (PS3) | [SHOW MORE] |
PS5 | -- | (CS1), (CS3), (CS7) | (PS4) | [SHOW MORE] |
Proving the conditions for criticality
Our goal is now to show that satisfies the conditions for being a critical subgroup. Recall that we have established that is a characteristic subgroup of and that . Also, recall that:
.
We check the conditions one by one:
- Frattini-in-center group condition: We need to show that : For this, observe that is contained in , which is elementary Abelian (it is contained in an Abelian group , and is generated by elements of order ). Thus, is elementary Abelian, and we get .
- Commutator-in-center subgroup condition: We need to show that : Modulo , the image of is contained in , which is contained in . Hence , which is trivial. Hence .
- Self-centralizing subgroup condition: We need to show that : We do this by contradiction. Since the proof is somewhat long, we do it in the table below.
ASSUMPTION: Suppose , such that is not in .
Step no. | Assertion | Facts used | Steps used (earlier parts) | Steps used (this part) | Contradiction-prone assumption used? | Explanation |
---|---|---|---|---|---|---|
PSC1 | is normal in | (Char1), (Norm3) | (PS3) | -- | no | [SHOW MORE] |
PSC2 | -- | (PS5) | -- | no | [SHOW MORE] | |
PSC3 | is nontrivial and normal in | (Norm1) | -- | (PSC1) | yes | [SHOW MORE] |
PSC4 | is trivial | -- | -- | (PSC2) | no | [SHOW MORE] |
PSC5 | intersects nontrivially | fact (1) about Omega-1 | -- | (PSC3) | yes | [SHOW MORE] |
PSC6 | , and thus, | -- | -- | -- | no | [SHOW MORE] |
PSC7 | -- | (CS3), (CS6),(CS7) | -- | no | [SHOW MORE] | |
PSC8 | is trivial | -- | -- | (PSC4), (PSC7) | no | [SHOW MORE] |
PSC9 | is nontrivial | -- | -- | (PSC5), (PSC6) | yes | [SHOW MORE] |
PSC10 | Contradiction! | -- | - | (PSC8) and (PSC9) | yes | [SHOW MORE] |
References
Journal references
- Solvability of groups of odd order by Walter Feit and John Griggs Thompson, Pacific Journal of Mathematics, Volume 13, Page 775 - 1029(Year 1963): This 255-page long paper gives a proof that odd-order implies solvable: any odd-order group (i.e., any finite group whose order is odd) is a solvable group.^{Project Euclid page}^{More info}: Thompson's critical subgroup theorem appears as Lemma 8.2, Section 8 (Miscellaneous preliminary lemmas) in Chapter 2, Page 795.
Textbook references
- Finite Groups by Daniel Gorenstein, ISBN 0821843427, Page 185, Theorem 3.11 (including Lemma 3.12, also theorem 3.13), Section 5.3, ^{More info}