Panferov Lie group
Definition
For a ring and a natural number less than or equal to its powering threshold
Suppose is a commutative unital ring and is a natural number greater than or equal to 2 such that the additive group of is powered over the set of all natural numbers strictly less than , or equivalently, is at most one more than the powering threshold of the additive group of .
The Panferov Lie group for and is defined as the Lazard Lie group (via the Lazard correspondence) for the Panferov Lie algebra (defined below) with parameters and .
For a prime number: the extreme case
Let be a prime number. The Panferov Lie group for is the Panferov Lie group corresponding to (the prime field) and .
Definition of Panferov Lie algebra being used
Suppose is a natural number greater than or equal to 2 and is a commutative unital ring. The Panferov Lie algebra of degree over the ring is a -Lie algebra (and hence also is a Lie ring) defined as follows:
- The additive group is a free module of rank with basis . Explicitly, the Lie algebra is
- The Lie bracket is defined as follows:
Particular cases
Powering thresholds for some ring choices
Choice of | Maximum permissible value of = one more than the powering threshold of (this is the smallest prime number such that is not powered over that number) |
---|---|
, the prime field for a prime number | |
, the field of rational numbers | no maximum, i.e., could be any natural number. |
where is a prime power that equals the size of the finite field and is the underlying prime (the field characteristic). | |
where is a prime number and is a positive integer. |
Panferov Lie groups for some small choices of
Description of the Panferov Lie group with parameter | Nilpotency class (equals ) | Derived length (equals ) | Theoretical maximum value of derived length for the nilpotency class (the theoretical maximum is ) | Is the derived length at the theoretical maximum for the nilpotency class? (No except for powers of 2 and Mersenne numbers, but otherwise off by one | |
---|---|---|---|---|---|
2 | additive group of . It is an abelian group. | 1 | 1 | 1 | Yes |
3 | unitriangular matrix group of degree three over . | 2 | 2 | 2 | Yes |
4 | Panferov Lie group of degree four | 3 | 2 | 2 | Yes |
5 | Panferov Lie group of degree five | 4 | 2 | 3 | No |
6 | Panferov Lie group of degree six | 5 | 2 | 3 | No |
7 | Panferov Lie group of degree seven | 6 | 3 | 3 | Yes |
The extreme case
Prime number | Panferov Lie group for | Order (equals ) | Nilpotency class (equals ) | Derived length (equals ) | Theoretical maximum value of derived length for the nilpotency class (the theoretical maximum is ) | Is the derived length at the theoretical maximum for the nilpotency class? (No except for and Mersenne primes, in general it'll be off by one but still close enough) |
---|---|---|---|---|---|---|
2 | Klein four-group | 4 | 1 | 1 | 1 | Yes |
3 | unitriangular matrix group:UT(3,3) | 27 | 2 | 2 | 2 | Yes |
5 | Panferov Lie group for 5 | 3125 | 4 | 2 | 3 | No |
7 | Panferov Lie group for 7 | 823543 | 6 | 3 | 3 | Yes |
Arithmetic functions
Function | Value in the general case | Value in the case | Explanation |
---|---|---|---|
order | same as the order of its Lazard Lie ring, which is -dimensional vector space over . | ||
exponent | characteristic of the ring | same as the exponent of its Lazard Lie ring, which is on account of it being a nonzero vector space over . | |
nilpotency class | same as the nilpotency class of its Lazard Lie ring; see Panferov Lie algebra#Arithmetic functions | ||
derived length | same as the derived length of its Lazard Lie ring, see Panferov Lie algebra#Arithmetic functions |
References
Journal references
- Nilpotent groups with lower central factors of minimal ranks by B. A. Panferov, Algebra and Logic, Volume 19, Page 455 - 459(Year 1980): ^{Gated copy (PDF)}^{More info}
Textbook references
- p-automorphisms of finite p-groups by Evgenii I. Khukhro, 13-digit ISBN 978-0-521-59717-3, 10-digit ISBN 0-521-59717-X, Page 126, Example 10.28, ^{More info}