Semidirect product of Z5 and Z8 via inverse map

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Definition

This group of order 40 can be defined by the following presentation (here, e is used for the identity element):

G := \langle a,b \mid a^5 = b^8 = e, bab^{-1} = a^{-1} \rangle

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 ]