General affine group:GA(1,9)

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Definition

This group is the general affine group of degree one over the field of nine elements. In other words, it is the semidirect product of the additive group of the field of nine elements and the multiplicative group of the field of nine elements.

Canonical matrix representation of elements

While any general affine group $GA(n,K)$ cannot be realized as a subgroup of the general linear group $GL(n,K)$, it can be realized as a subgroup of $GL(n+1,K)$ in a fairly typical way: the vector from $K^{n}$ is the first $n$ entries of the right column, the matrix from $GL(n,K)$ is the top left $n\times n$ block, there is a $1$ in the bottom right corner, and zeroes elsewhere on the bottom row. In particular, $GA(1,9)$ is the set of matrices over $\mathbb {F} _{9}$ of the form ${\begin{pmatrix}a&b\\0&1\end{pmatrix}}$ with $a\neq 0$.

Arithmetic functions

Function	Value	Explanation
order	72
exponent	24
Frattini length	1
Fitting length	2
minimum size of generating set	2
subgroup rank	2

Group properties

Property	Satisfied	Explanation
abelian group	No
nilpotent group	No
supersolvable group	No
solvable group	Yes
Frobenius group	Yes

GAP implementation

Group ID

This finite group has order 72 and has ID 39 among the groups of order 72 in GAP's SmallGroup library. For context, there are groups of order 72. It can thus be defined using GAP's SmallGroup function as:

SmallGroup(72,39)

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

gap> G := SmallGroup(72,39);

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

IdGroup(G) = [72,39]

or just do:

IdGroup(G)

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