General affine group:GA(1,5)

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$GA(1,5)$ is a certain non-abelian group of order 20.

Definition

This group is defined in the following equivalent ways:

1. It is the general affine group of degree one over the field of five elements. That is, it is the set of affine transformations $x\mapsto ax+b$ for $a\in \mathbb {F} _{5}\backslash \{0\}$, $b,x\in \mathbb {F} _{5}$ under the group operation of composition.
2. It is the semidirect product of the additive and multiplicative groups of the field of five elements.
3. It is the holomorph of the cyclic group of order five. That is, it is a semidirect product of cyclic group:Z5 with cyclic group:Z4.
4. It is the Suzuki group $Sz(2)$ or the Suzuki group $Sz(2^{1+2m})$ where $m=0$. Note: This is the only non-simple Suzuki group.
5. It is the Galois group of $X^{5}-a$ where $a$ is an element of $\mathbb {Q}$ which isn’t a fifth power.

The group can be given by the presentation, with $e$ denoting the identity element:

$\langle a,b\mid a^{5}=b^{4}=e,bab^{-1}=a^{2}\rangle$

It is denoted $GA(1,5)$.

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, $GA(1,5)$ is the set of matrices over $\mathbb {F} _{5}$ of the form

${\begin{pmatrix}a&b\\0&1\end{pmatrix}}$

with $a\neq 0$.

Arithmetic functions

Function	Value	Similar groups	Explanation
order (number of elements, equivalently, cardinality or size of underlying set)	20	groups with same order	As $GA(1,q)$, $q=5$: $q(q-1)=5(5-1)=20$ As holomorph of cyclic group:Z5: $|\mathbb {Z} _{5}||\operatorname {Aut} (\mathbb {Z} _{5})|=5\cdot 4=20$ As $\!Sz(q),q=2$: $q^{2}(q^{2}+1)(q-1)=2^{2}(2^{2}+1)(2-1)=4\cdot 5\cdot 1=20$
exponent of a group	20	groups with same order and exponent of a group | groups with same exponent of a group	The subgroup of order five has an element of order five; the acting subgroup of order four is cyclic and has an element of order four.
Frattini length	1	groups with same order and Frattini length | groups with same Frattini length
Fitting length	2	groups with same order and Fitting length | groups with same Fitting length
derived length	2	groups with same order and derived length | groups with same derived length
minimum size of generating set	2	groups with same order and minimum size of generating set | groups with same minimum size of generating set	Generator of cyclic group of order five, generator of acting cyclic group of order four. $\langle (1,2,3,4,5),(2,3,5,4)\rangle$ works as a permutation representation.
subgroup rank of a group	2	groups with same order and subgroup rank of a group | groups with same subgroup rank of a group

Group properties

Function	Value	Explanation
abelian group	No
nilpotent group	No
metacyclic group	Yes
supersolvable group	Yes
solvable group	Yes
Frobenius group	Yes
Camina group	Yes

Linear representation theory

Further information: Linear representation theory of general affine group:GA(1,5)

Elements

Further information: Element structure of general affine group:GA(1,5)

Orders

$GA(1,5)$ has elements of the following orders:

order	number of elements with that order
1	1
2	5
4	10
5	4

Conjugacy classes

$GA(1,5)$ has 5 conjugacy classes.

Subgroups

Quick summary

Item	Value
Number of subgroups	14
normal subgroups	There are four normal subgroups: the whole group, D10 in GA(1,5), Z5 in GA(1,5), and the trivial subgroup.

Permutation representation

An example of a permutation representation of this group: $GA(1,5)$ is isomorphic to the subgroup of symmetric group:S5 given by $\langle (1,2,3,4,5),(2,3,5,4)\rangle$.

GAP implementation

Group ID

This finite group has order 20 and has ID 3 among the groups of order 20 in GAP's SmallGroup library. For context, there are groups of order 20. It can thus be defined using GAP's SmallGroup function as:

SmallGroup(20,3)

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

gap> G := SmallGroup(20,3);

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

IdGroup(G) = [20,3]

or just do:

IdGroup(G)

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