## Cayley Tables

Group Multiplication Tables

The Cayley table for \(S_3\)

\(S_3\) is the symmetric group on three elements. Every element in this group can be represented in cycle notation. In cycle notation \((a\ b\ c)\) means:
\[ a \rightarrow b \rightarrow c \rightarrow a \]
Notice how \(c\) "cycles" around and maps to the first element \(a\). If a number is not in the cycle that means it maps to itself. \[\\\]
Here is your assignment: verify this Cayley table. It's \(36\) permutation multiplications, and verifying this will sharpen your skills.

Kernel of Group Homomorphisms

When working with small groups, especially when you are starting out in abstract algebra, it can be helping to create a table with all possible multiplications.