Semidirect product of Z5 and Z8 via inverse map

This article is about a particular group, i.e., a group unique upto isomorphism. View specific information (such as linear representation theory, subgroup structure) about this group
View a complete list of particular groups (this is a very huge list!)[SHOW MORE]

VIEW FACTS ABOUT THIS GROUP: All factsGroup property satisfactionsSubgroup property satisfactionsGroup property dissatisfactionsSubgroup property satisfactions
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 of order 40 can be defined by the following presentation (here, $e$ is used for the identity element):

$G:=⟨a,b\mid a^5=b^8=e,ba{b}^{-1}=a^{-1}⟩$

GAP implementation

Group ID

This finite group has order 40 and has ID 1 among the groups of order 40 in GAP's SmallGroup library. For context, there are groups of order 40. It can thus be defined using GAP's SmallGroup function as:

SmallGroup(40,1)

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

gap> G := SmallGroup(40,1);

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

IdGroup(G) = [40,1]

or just do:

IdGroup(G)

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

Description by presentation

The group can be defined in GAP using its presentation as follows:

gap> F := FreeGroup(2);
<free group on the generators [ f1, f2 ]>
gap> G := F/[F.1^5,F.2^8,F.2*F.1*F.2^(-1)*F.1];
<fp group on the generators [ f1, f2 ]>

We can check that this is the correct group:

gap> IdGroup(G);
[ 40, 1 ]