Panferov Lie group for 7

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VIEW DEFINITIONS USING THIS AS EXAMPLE: All terms |Group properties |Subgroup properties|
VIEW FACTS USING THIS AS EXAMPLE: All facts |Group property non-implications |Subgroup property non-implications |Group metaproperty dissatisfactionsSubgroup metaproperty dissatisfactions

Definition

This group is defined as the Panferov Lie group for the prime number 7. Explicitly, it is the Lazard Lie group corresponding to the Panferov Lie algebra for the prime 7. The definition of the Panferov Lie algebra for a prime number is below:

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:

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

Arithmetic functions

Want to compare and contrast arithmetic function values with other groups of the same order? Check out groups of order 3125#Arithmetic functions

Function	Value	Similar groups	Explanation
order (number of elements, equivalently, cardinality or size of underlying set)	823543	groups with same order
prime-base logarithm of order	7	groups with same prime-base logarithm of order
max-length of a group	7		max-length of a group equals prime-base logarithm of order for group of prime power order
chief length	7		chief length equals prime-base logarithm of order for group of prime power order
composition length	7		composition length equals prime-base logarithm of order for group of prime power order
exponent of a group	7	groups with same order and exponent of a group | groups with same exponent of a group
prime-base logarithm of exponent	1	groups with same order and prime-base logarithm of exponent | groups with same prime-base logarithm of order and prime-base logarithm of exponent | groups with same prime-base logarithm of exponent
nilpotency class	6	groups with same order and nilpotency class | groups with same prime-base logarithm of order and nilpotency class | groups with same nilpotency class	As Panferov Lie group for $p$, $p=5$, $\lceil \log _{2}(p+2)-1\rceil =\lceil \log _{2}(7)-1\rceil =2$. See Panferov Lie group#Arithmetic functions.
derived length	3	groups with same order and derived length | groups with same prime-base logarithm of order and derived length | groups with same derived length	As Panferov Lie group for $p$, $p=7$, $\lceil \log _{2}(p+2)-1\rceil =\lceil \log _{2}(9)-1\rceil =3$. See Panferov Lie group#Arithmetic functions.