How to write the the Cayley table of the group given by the following Cayley diagram ?
Answr:
From the diagram I understand that the group contains $ \ 10 \ $ elements and the group is generated by two elements .
The group should be $ \{<a,b> | a^2=1, b^5=1 \} \ $
The elements should be $ \ \{e,b,b^2,b^3,b^4,a,ab,ab^2,ab^3,ab^4 \} \ $ .
But what should be the multiplication rule between the elements ?
I actually am unable to write the table ?
Can anyone help me with the Cayley table ?
This is $D_{10}$, the dihedral group of order $10$.
The presentation is $\langle a,b \mid a^2=b^5=1, (ab)^2=1 \rangle$.
Note that your presentation would be infinite.
The Cayley table:
$$\begin{array}{l|l} \cdot & e & b & b^2 & b^3 & b^4 & a & ab & ab^2 & ab^3 & ab^4 \\\hline e & e & b & b^2 & b^3 & b^4 & a & ab & ab^2 & ab^3 & ab^4 \\\hline b & b & b^2 & b^3 & b^4 & e & ab^4 & a & ab & ab^2 & ab^3 \\\hline b^2 & b^2 & b^3 & b^4 & e & b & ab^3 & ab^4 & a & ab & ab^2 \\\hline b^3 & b^3 & b^4 & e & b & b^2 & ab^2 & ab^3 & ab^4 & a & ab \\\hline b^4 & b^4 & e & b & b^2 & b^3 & ab & ab^2 & ab^3 & ab^4 & a \\\hline a & a & ab & ab^2 & ab^3 & ab^4 & e & b & b^2 & b^3 & b^4 \\\hline ab & ab & ab^2 & ab^3 & ab^4 & a & b^4 & e & b & b^2 & b^3 \\\hline ab^2 & ab^2 & ab^3 & ab^4 & a & ab & b^3 & b^4 & e & b & b^2 \\\hline ab^3 & ab^3 & ab^4 & a & ab & ab^2 & b^2 & b^3 & b^4 & e & b \\\hline ab^4 & ab^4 & a & ab & ab^2 & ab^3 & b & b^2 & b^3 & b^4 & e \end{array}$$