Quick Checks

Tips to help as you get started

  • Distinct Rows / Columns

    No two rows or columns will be the same. If the row or column for \(x\) and \(y\) were the same, then \[x*z=y*z\] for any \(z\). Multiply both sides by \(z^{-1}\) to see that \(x=y\). A contradiction!

  • No Repeats

    Each row and column of the Cayley Table should have all the group elements in some order - no element should be repeated. To prove this, imagine there were repeated elements in a row. Can you find the contradiction?

Example

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.
The Cayley table for the symmetric group S3

Free Course

Created by Socratica

Related Lesson

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.
#LearnMore
A line and a circle glowing