General semilinear group:GammaL(1,8)

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Definition

This group is defined in the following equivalent ways:

1. It is the general semilinear group of degree one over field:F8.
2. It is the external semidirect product of cyclic group:Z7 by cyclic group:Z3 for the unique nontrivial action of the latter on the former.

Properties

The group $\mathbb {Z} _{7}\rtimes \mathbb {Z} _{3}$ is the smallest non-abelian group of odd order. It is one of the two groups of order 21 alongside cyclic group:Z21.

This group is a soluble group.

This group is a Frobenius group of order 21.

Construction as a semidirect product

We will construct this group as a semidirect product $\mathbb {Z} _{7}\rtimes \mathbb {Z} _{3}$.

We need to find a group homomorphism $\rho :\mathbb {Z} _{3}\to \operatorname {Aut} (\mathbb {Z} _{7})\cong \mathbb {Z} _{6}$.

Consider the map $x\mapsto (y\mapsto 2^{x}y)$

Here, $x$ is an element of the integers mod $3$, and $y$ is an element of the integers mod $7$.

This is a homomorphism, since $\rho (xx')=y\mapsto 2^{xx'}y=(y\mapsto 2^{x}y)\circ (y\mapsto 2^{x'}y)=\rho (x)\rho (x')$. Indeed, $\rho (1)=y\mapsto y$, the identity element of $\mathrm {Aut} (\mathbb {Z} _{7})$.

Then $G=\mathbb {Z} _{7}\rtimes _{\rho }\mathbb {Z} _{3}$ is non-abelian since $(0,1)*(1,0)=(2,1)$, $(1,0)*(0,1)=(1,1)$.

Hence we have constructed a non-abelian group of order $21$.

The classification of groups of order 21 says that there is only one non-abelian group of order $21$, the Frobenius group, hence this is the Frobenius group.

Arithmetic functions

Basic arithmetic functions

Function	Value	Similar groups	Explanation
order (number of elements, equivalently, cardinality or size of underlying set)	21	groups with same order	As $\Gamma L(1,q),q=p^{r},q=8,p=2,r=3$: $r(q-1)=3(8-1)=3(7)=21$ As semidirect product of groups of order 7 and 3: $7\times 3=21$ (see order of semidirect product is product of orders)
exponent of a group	21	groups with same order and exponent of a group | groups with same exponent of a group	There are elements of order 7 and 3
nilpotency class	--	--	not a nilpotent group
derived length	2	groups with same order and derived length | groups with same derived length	Derived subgroup is isomorphic to cyclic group:Z7.
Fitting length	2	groups with same order and Fitting length | groups with same Fitting length	Fitting subgroup is same as derived subgroup
Frattini length	1	groups with same order and Frattini length | groups with same Frattini length	Maximal subgroups of orders 7 and 3 intersect trivially
minimum size of generating set	2	groups with same order and minimum size of generating set | groups with same minimum size of generating set	Not cyclic, can be generated by two elements since semidirect product of cyclic group:Z3 and cyclic group:Z7, hence two.

Arithmetic functions of a counting nature

Function	Value	Similar groups	Explanation
number of conjugacy classes	5	groups with same order and number of conjugacy classes | groups with same number of conjugacy classes	As $\Gamma L(1,p^{3}),p=2$: $(p^{3}+8p-9)/3=(2^{3}+8\cdot 2-9)/3=5$ See element structure of general semilinear group of degree one over a finite field

GAP implementation

Group ID

This finite group has order 21 and has ID 1 among the groups of order 21 in GAP's SmallGroup library. For context, there are groups of order 21. It can thus be defined using GAP's SmallGroup function as:

SmallGroup(21,1)

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

gap> G := SmallGroup(21,1);

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

IdGroup(G) = [21,1]

or just do:

IdGroup(G)

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