Thompson's replacement theorem for abelian subgroups
This article defines a replacement theorem
View a complete list of replacement theorems| View a complete list of failures of replacement
Contents
Statement
Suppose is a group of prime power order (we'll call the prime ).
Let denote the set of all abelian subgroups of maximum order in (i.e., for all abelian subgroups of ).
Suppose and is an abelian subgroup such that normalizes but does not normalize . Then, there exists an abelian subgroup of such that:
- .
- is a proper subgroup of .
- is contained in , i.e., it is contained in the normal closure of in . In particular, it is contained in the normal closure of in .
- normalizes .
We have the following additional conclusions:
- normalizes : Note that since is contained in , and by assumption, we see that normalizes as well.
- is a proper subgroup of : This is because cannot contain , since normalizes but does not normalize .
Thus, all the conditions assumed for also hold for , except possibly the fact that does not normalize . Hence, the term replacement.
Related facts
Similar replacement theorems
- Thompson's replacement theorem for abelian subgroups in arbitrary finite groups
- Thompson's replacement theorem for elementary abelian subgroups
- Glauberman's replacement theorem
- Timmesfeld's replacement theorem
For a complete list of replacement theorems, refer:
Applications
- Stable version of Thompson's replacement theorem for abelian subgroups: This is a result obtained via repeated application of Thompson's replacement theorem as long as does not normalize the (replaced) .
- Any abelian normal subgroup normalizes an abelian subgroup of maximum order
- There exists an abelian subgroup of maximum order whose normalizer contains every abelian subgroup it normalizes
Facts used
- Thompson's lemma on product with centralizer of commutator with abelian subgroup of maximum order
- Nilpotent implies center is normality-large
- Normality satisfies image condition
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 . is the set of abelian subgroups of of maximum order. , and is a subgroup of such that normalizes but does not normalize .
To prove: There exists such that is a proper subgroup of , normalizes and , and is contained in .
Proof: We let and .
Step no. | Assertion | Facts used | Given data used | Proof steps used | Explanation |
---|---|---|---|---|---|
1 | is a subgroup of , and is normal in | -- | normalizes | -- | [SHOW MORE] |
2 | normalizes | -- | is abelian | -- | [SHOW MORE] |
3 | normalizes | -- | normalizes | -- | [SHOW MORE] |
4 | is normal in | -- | -- | steps (2), (3) | [SHOW MORE] |
5 | is a proper subgroup of | -- | does not normalize | [SHOW MORE] | |
6 | is a nontrivial normal subgroup of | fact (3) | -- | steps (1), (4), (5) | [SHOW MORE] |
7 | is nontrivial | fact (2) | -- | step (6) | Direct combination of the facts |
8 | Let be such that its image mod is a nontrivial element in . Then is contained in and hence in | step (7) | Note that step (7) guarantees the existence of such . [SHOW MORE] | ||
9 | is an abelian subgroup of | is abelian | step (8) | [SHOW MORE] | |
10 | is an abelian subgroup of maximum order in | fact (1) | is abelian of maximum order | step (9) | Piece together. |
11 | is contained in | is abelian | step (8) | [SHOW MORE] | |
12 | is contained in | -- | step (11) | [SHOW MORE] | |
13 | normalizes | step (8) | [SHOW MORE] | ||
14 | is not contained in | step (8) | [SHOW MORE] | ||
15 | is not contained in | steps (8), (14) | [SHOW MORE] | ||
16 | is contained in | steps (1), (8), (10) | [SHOW MORE] | ||
17 | is contained in . | steps (8), (9), (10) | [SHOW MORE] | ||
18 | normalizes | step (1), (16) | [SHOW MORE] |
The conclusions of steps (10), (12), (13), (15), (16), (17), and (18) complete the proof.
Conceptual summary of proof
The key idea of the proof is to use fact (1) somehow, and this requires finding a suitable such that is abelian. and the resulting subgroup that we get is nice enough.
Working backward from this, we see that we would like that not be a subgroup of , which in turn means that we want outside . On the other hand, we do want to normalize , which means that we want to lie inside . Finally, we would like so that the subsequent group constructed is in . With all these considerations, we should aim to find outside but inside , but such that .
This motivation now allows us to work out the proof details given above. Here is a rough outline of the proof:
- Steps (1)-(6) set things up. The fact that does not normalize is used to show that is a nontrivial normal subgroup of .
- The crucial steps for the construction are (7) and (8), that involve the choice of . Steps (9) and (10) now use fact (1) to get a hold on .
- The remaining steps help us get a clearer idea of the subgroup inclusions and prove the remaining desired properties for .
GAP implementation
Here is a GAP implementation of the constructive approach used in the proof:
ThompsonsReplacementAbelian := function(P,A,B) local C,N,x,M,CAM,Astar; C := Group(Union(A,B)); N := Normalizer(B,A); x := Filtered(Set(B),y -> (not(y in N)) and IsSubgroup(N,CommutatorSubgroup(Group(y),C)))[1]; M := Group(List(Set(A),y -> x^(-1) * y^(-1) * x * y)); CAM := Centralizer(A,M); Astar := Group(Union(M,CAM)); return Astar; end;;
References
Textbook references
- Finite Groups by Daniel Gorenstein, ISBN 0821843427, Page 273, Theorem 2.5, ^{More info}
Journal references
- A replacement theorem for p-groups and a conjecture by John Griggs Thompson, Journal of Algebra, ISSN 00218693, Volume 13, Page 149 - 151(Year 1969): In this paper, Thompson proved the replacement theorem for abelian subgroups, as well as the result that group of prime power order having a larger abelianization than any proper subgroup has class two.^{Official copy}^{More info}
- An extension of Thompson's replacement theorem by algebraic group methods by George Glauberman, Finite Groups 2003