Elementary abelian-to-normal replacement theorem for large primes
This article defines a replacement theorem
View a complete list of replacement theorems| View a complete list of failures of replacement
Contents
- 1 Statement
- 2 Related facts
- 3 Facts used
- 4 Proof
- 4.1 Overall proof strategy
- 4.2 Base case for induction
- 4.3 Inductive step
- 4.4 Part one: reduction to the case that is 2-subnormal and its normal closure has exponent
- 4.5 Part two: reduction to the case where there is a self-centralizing (in ) elementary abelian normal (in ) subgroup of order
- 4.6 Part three: the final reduction to fact (1)
- 5 References
Statement
Hands-on statement
Let be a group of prime power order, say , and let be an elementary abelian subgroup of of order .
Suppose , and is odd.
Then, there exists an elementary abelian normal subgroup of satisfying:
- has order (same as )
- is contained in the normal closure of
Statement in terms of normal replacement conditions
Suppose is a natural number and is an odd prime. Then, the singleton collection of the elementary abelian group of order is a Collection of groups satisfying a strong normal replacement condition (?). In particular, it is also a Collection of groups satisfying a weak normal replacement condition (?).
Statement/corollary in terms of normal rank
- For an odd prime , a -group whose rank is at most has the property that its rank equals its normal rank.
- For an odd prime , a -group whose normal rank is at most has the property that its rank equals its normal rank.
Related facts
Note that this fact is superseded by the combination of the following two facts:
- Glauberman's abelian-to-normal replacement theorem for bounded exponent and half of prime plus one
- Jonah-Konvisser abelian-to-normal replacement theorem
Facts used
Main fact used
- Abelian-to-normal replacement theorem for prime exponent, which is Theorem B of the same paper. It states the following: Suppose is a finite Group of prime exponent (?): group of prime power order, say , and with exponent (so every element has order ). Suppose is an abelian subgroup of order , and nilpotency class at most .
Then, there exists an Abelian normal subgroup (?) of such that:
- is contained in the normal closure of in
- has the same order (i.e., ) as
Other facts used
Proof
The proof given here is (largely) the same as that presented in the original paper where the theorem appeared.
Overall proof strategy
We fix the prime beforehand. The overall proof strategy is to use a double induction, first on and then on . Note that the variables and differ in one important respect: is bounded from above in terms of , so that induction proceeds only for finitely many steps for any fixed (though the number of steps increases as increases), whereas is not bounded.
By double induction, we mean that in order to prove the statement for a particular pair , we assume the statement to be true for where and and for where .
We use some simple reasoning to ultimately reduce to the case of Fact (1), which is Theorem A of the same paper.
Base case for induction
This case is obvious. PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE]Inductive step
Given: A group of order , prime. An elementary abelian subgroup of of order , with .
To prove: There exists a normal subgroup of of order such that is contained in the normal closure of in .
Part one: reduction to the case that is 2-subnormal and its normal closure has exponent
This part uses the inductive hypothesis on .
To prove: It suffices to consider the case where is 2-subnormal and the normal closure of in has exponent and nilpotency class at most , which is less than .
Proof: We will show that there is a subgroup lying in the normal closure of such that is 2-subnormal and its normal closure has exponent and nilpotency class at most , which is less than . In particular, Thus, if we show that the result holds for all groups satisfying the conditions established for , we get the result for and hence for .
Step no. | Assertion | Facts used | Given data/previous part steps used | Steps used | Explanation |
---|---|---|---|---|---|
PA1 | If , we are done. Hence, we can restrict henceforth to the case , so is proper in . | is normal in itself, hence is normal and we are done. | |||
PA2 | There exists a maximal subgroup of containing . is normal in and has order . In particular, contains the normal closure of in . | (O1) | Step (PA1): proper in | ||
PA3 | There exists an elementary abelian normal subgroup of of order . | Inductive hypothesis on . | Step (PA2): contains . | [SHOW MORE] |
Steps (PA4), (PA5), and (PA6) establish the existence of contained in satisfying all the required properties.
Part two: reduction to the case where there is a self-centralizing (in ) elementary abelian normal (in ) subgroup of order
This part uses the inductive hypothesis on . Note that we assume that satisfies the properties established for in part one.
To prove: It suffices to consider the case where is 2-subnormal and there is an elementary abelian normal subgroup of of order and inside such that .
Proof:
Step no. | Assertion | Facts used | Given data/previous part steps used | Steps used | Explanation |
---|---|---|---|---|---|
PB1 | contains an elementary abelian subgroup of order . | An elementary abelian subgroup of order contains elementary abelian subgroups of all orders . | has order . | ||
PB2 | There exists an elementary abelian normal subgroup of lying inside of order . | Inductive hypothesis on | (PB1) | [SHOW MORE] | |
PB3 | If , then we can find an abelian normal subgroup of containing and of order such that | (O1), (O4)-(O8) | -- | (PB2) | [SHOW MORE] |
PB4 | The constructed in (PB3), subject to the condition that , is elementary abelian | -- | has exponent | (PB3) | <toggledisplay>Since has exponent , the subgroup is abelian of exponent , hence is elementary abelian. |
PB5 | It suffices to consider the case . | -- | -- | (PB3), (PB4) | By the previous two steps, if , then works as a replacement for . Hence, it suffices to restrict attention to the case that . |
Note that the lengthy argument in (PB3) is similar to the proof that maximal among abelian normal implies self-centralizing in nilpotent. The main difference is that here, normality is in but the self-centralizing subgroup condition is in the smaller subgroup . Thus, although the proof technique remains the same, we cannot directly use the other statement as a black box.
Similar arguments to (PB3) are also used in the proof of Thompson's replacement theorem for abelian subgroups and Thompson's critical subgroup theorem.
Part three: the final reduction to fact (1)
PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE]References
- Limits of abelian subgroups of finite p-groups by Jonathan Lazare Alperin and George Isaac Glauberman, Journal of Algebra, ISSN 00218693, Volume 203, Page 533 - 566(Year 1998): ^{Weblink for Elsevier copy}^{More info}, Theorem A