Cayley Tables: Structural Representation of Finite Groups

1. Cayley Tables are used for Structural Representation of Finite Groups. A Cayley Table Lists all the Elements of a Finite Group and Results of Group Operation between all Possible Pair of Elements of the Group.
2. Following gives the Cayley Table Representation of any Finite Group \(G\) having \(N\) Elements \(e_1, e_2, \cdots, e_N\) having the Group Operation Represented by Operator \(\circ\)
   
   

   \(\circ\)	\(e_1\)	\(e_2\)	\(e_3\)	\(\cdots\)	\(e_N\)
   \(e_1\)	\(e_1 \circ e_1\)	\(e_1 \circ e_2\)	\(e_1 \circ e_3\)	\(\cdots\)	\(e_1 \circ e_N\)
   \(e_2\)	\(e_2 \circ e_1\)	\(e_2 \circ e_2\)	\(e_2 \circ e_3\)	\(\cdots\)	\(e_2 \circ e_N\)
   \(e_3\)	\(e_3 \circ e_1\)	\(e_3 \circ e_2\)	\(e_3 \circ e_3\)	\(\cdots\)	\(e_2 \circ e_N\)
   \(\vdots\)	\(\vdots\)	\(\vdots\)	\(\vdots\)	\(\ddots\)	\(\vdots\)
   \(e_N\)	\(e_N \circ e_1\)	\(e_N \circ e_2\)	\(e_N \circ e_3\)	\(\cdots\)	\(e_N \circ e_N\)

   
   As a convention, the First Element given in any Cayley Table is the Identity Element of the Group that it represents.
   
   In an Actual Cayley Table, any Operation (for example \(e_1 \circ e_2\)) is given by the Actual Result of the Operation.
3. The following example gives the Cayley Table Representation of the Group \(G=\{1,i,-1,-i\}\) under the Binary Operation of Multiplication represented by the Operator \(\times\)
   
   

   \(\times\)	\(1\)	\(i\)	\(-1\)	\(-i\)
   \(1\)	\(1\)	\(i\)	\(-1\)	\(-i\)
   \(i\)	\(i\)	\(-1\)	\(-i\)	\(1\)
   \(-1\)	\(-1\)	\(-i\)	\(1\)	\(i\)
   \(-i\)	\(-i\)	\(1\)	\(i\)	\(-1\)

   
   In the Group \(G\) and Cayley Table above, \(i=\sqrt{-1}\).