Panferov Lie group

Definition

For a ring and a natural number less than or equal to its powering threshold

Suppose $R$ is a commutative unital ring and $n$ is a natural number greater than or equal to 2 such that the additive group of $R$ is powered over the set of all natural numbers strictly less than $n$, or equivalently, $n$ is at most one more than the powering threshold of the additive group of $R$.

The Panferov Lie group for $R$ and $n$ is defined as the Lazard Lie group (via the Lazard correspondence) for the Panferov Lie algebra (defined below) with parameters $R$ and $n$.

For a prime number: the extreme case

Let $p$ be a prime number. The Panferov Lie group for $p$ is the Panferov Lie group corresponding to $R=\mathbb {F} _{p}$ (the prime field) and $n=p$.

Definition of Panferov Lie algebra being used

Suppose $n$ is a natural number greater than or equal to 2 and $R$ is a commutative unital ring. The Panferov Lie algebra of degree $n$ over the ring $R$ is a $R$-Lie algebra (and hence also is a Lie ring) defined as follows:

• The additive group is a free module of rank $n$ with basis $e_{1},e_{2},\dots ,e_{n}$. Explicitly, the Lie algebra is $\bigoplus _{i=1}^{n}Re_{i}$
• The Lie bracket is defined as follows:

$[e_{i},e_{j}]=\left\lbrace {\begin{array}{rl}(i-j)e_{i+j},&i+j\leq n\\0,&i+j>n\\\end{array}}\right.$

Particular cases

Powering thresholds for some ring choices

Choice of $R$	Maximum permissible value of $n$ = one more than the powering threshold of $R$ (this is the smallest prime number such that $R$ is not powered over that number)
$R=\mathbb {F} _{p}$, the prime field for a prime number $p$	$p$
$R=\mathbb {Q}$, the field of rational numbers	no maximum, i.e., $n$ could be any natural number.
$R=\mathbb {F} _{q}$ where $q$ is a prime power that equals the size of the finite field $\mathbb {F} _{q}$ and $p$ is the underlying prime (the field characteristic).	$p$
$R=\mathbb {Z} /p^{k}\mathbb {Z}$ where $p$ is a prime number and $k$ is a positive integer.	$p$

The extreme case

Prime number $p$	Panferov Lie group for $p$	Order	Nilpotency class (equals $p-1$)	Derived length (equals $\lceil \log _{2}(p+2)-1\rceil$)	Theoretical maximum value of derived length for the nilpotency class (the theoretical maximum is $\lfloor \log _{2}(p-1)+1\rfloor$)	Is the derived length at the theoretical maximum for the nilpotency class? (No except for $p=2$ 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 $R=\mathbb {F} _{p},n=p$	Explanation
order	$|R|^{n}$	$p^{p}$	same as the order of its Lazard Lie ring, which is $p$-dimensional vector space over $\mathbb {F} _{p}$.
exponent	characteristic of the ring $R$	$p$	same as the exponent of its Lazard Lie ring, which is $p$ on account of it being a nonzero vector space over $\mathbb {F} _{p}$.
nilpotency class	$n-1$	$p-1$	same as the nilpotency class of its Lazard Lie ring; see Panferov Lie algebra#Arithmetic functions
derived length	$\lceil \log _{2}(n+2)-1\rceil$	$\lceil \log _{2}(p+2)-1\rceil$	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}