# Panferov Lie group for 5

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## Definition

This group is defined as the Panferov Lie group for the prime number 5. Explicitly, it is the Panferov Lie group where the ring is field:F5 and $n = 5$.

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:

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

## 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

## GAP implementation

### Group ID

This finite group has order 3125 and has ID 33 among the groups of order 3125 in GAP's SmallGroup library. For context, there are 77 groups of order 3125. It can thus be defined using GAP's SmallGroup function as:

SmallGroup(3125,33)

For instance, we can use the following assignment in GAP to create the group and name it $G$:

gap> G := SmallGroup(3125,33);

Conversely, to check whether a given group $G$ is in fact the group we want, we can use GAP's IdGroup function:

IdGroup(G) = [3125,33]

or just do:

IdGroup(G)

to have GAP output the group ID, that we can then compare to what we want.