# Question regarding Cayley graph generated in GAP software

I have computed a semidirect product of $$(\mathbb{Z}_5 \times \mathbb{Z}_5) \rtimes \mathbb{Z}_3$$ in GAP as shown below.

gap> Z3:=Group((1,2,3));;

gap>  Z3gen:=GeneratorsOfGroup(Z3)[1];;

gap>  Z5:=Group((1,2,3,4,5));;

gap> h:=DirectProduct(Z5,Z5);;

gap>  Auts:=AutomorphismGroup(h);;

gap> AutsOfOrd3:=Filtered(Elements(Auts),elt->Order(elt)=3);;

gap> ExHom:=GroupHomomorphismByImages(Z3,Auts,[Z3gen],[AutsOfOrd3[1]]);
[ (1,2,3) ] -> [ [ (1,2,3,4,5), (6,7,8,9,10) ] -> [ (1,5,4,3,2)(6,10,9,8,7), (1,2,3,4,5) ] ]

gap> s:=SemidirectProduct(Z3,ExHom,h);;


After that I drew a Cayley graph for the semidirect product $$s$$.

gap> Read("C:/Users/kasuni/Desktop/Test1/UndirectedGeneratingSets.gap");

gap> S:=IrredUndirGenSetsUpToAut(s);;

gap> S1:=S[1];;

gap> X1:=CayleyGraph(s,S1);


In the manual for grape package https://www.gap-system.org/Manuals/pkg/grape/doc/manual.pdf in page 12 it was described as

"if undirected=true (the default) then vertices $$x, y$$ are joined by an edge if and only if there is a $$g$$ in the list gens with $$y = gx$$ or $$y = g^{−1}x$$; if undirected=false then $$[x, y]$$ is an edge if and only if there is a $$g$$ in gens with $$y = gx$$." under the description for Cayley graphs. There I observed that the multiplication by the elements from the generating set is from the left side.

But according to the usual definition for Cayley graphs,

"Let $$G$$ be a group and $$S \subseteq G$$ be a generating set of $$G$$. The Cayley digraph of $$G$$ with respect to $$S$$, $$X=\overrightarrow{\operatorname{Cay}}(G,\: S)$$ is a graph whose vertices are the elements of $$G$$ and there is an edge from $$g$$ to $$gs$$ whenever $$g \in G$$ and $$s \in S$$. The Cayley graph, $$X=\operatorname{Cay}(G,\: S)$$ is the undirected graph whose vertices are the elements of $$G$$ and there is an edge from $$g$$ to $$gs$$ and from $$g$$ to $$gs^{-1}$$ whenever $$g \in G$$ and $$s \in S$$."

the multiplication by the elements from the generating set is on the right side.

Does that mean that the Cayley graph generated by the GAP software is different from the Cayley graph we think of under usual definitions?

If I try to draw a Cayley graph by computing every and each point by thinking about the usual definition where there is an edge from $$g$$ to $$gs$$ whenever $$g \in G$$ and $$s \in S$$, and do computations in GAP as,

A:=Elements(s);
P1:=A[1]*S1[1];
P2:=P1*S1[1];
P3:=P2*S1[1];
P4:=A[1]*S1[2];
P5:=P4*S1[2]; etc.


will it generate a Cayley graph corresponding to the usual definition?

What is its relationship with the Cayley graph given by GAP system's CayleyGraph command?

Can someone please help me to clarify this question regarding the Cayley graphs?

Thanks a lot in advance.

• I suggest to replace one of the tags by gap so people watching the gap tag will discover this (I have math.stackexchange.com/search?tab=active&q=gap in my bookmarks but others may not). Aug 21, 2019 at 8:47
• Thank you very much @AlexanderKonovalov . Please be kind enough to advice me with regarding to my question as well, if possible :) Aug 21, 2019 at 8:50
• Short of time right now, but hopefully someone will notice the question and explain. Aug 21, 2019 at 9:12