Fill in the Cayley table for S3 using the elements listed in...

Fill in the Cayley table for $S_3$ using the elements listed in the first row or column, and show that $S_3$ is non-abelian. $$ \begin{array}{l||l|l|l|l|l|l} \circ & e & \sigma & \sigma^2 & \tau & \sigma \tau & \sigma^2 \tau \\ \hline \hline e & & & & & & \\ \hline \sigma & & & & & & \\ \hline \sigma^2 & & & & & & \\ \hline \tau & & & & & & \\ \hline \sigma \tau & & & & & & \\ \hline \sigma^2 \tau & & & & & & \\ \hline \end{array} $$

Fill in the Cayley table for $S_3$ using the elements listed in the first row or column, and show that $S_3$ is non-abelian.
$$
\begin{array}{l||l|l|l|l|l|l}
\circ & e & \sigma & \sigma^2 & \tau & \sigma \tau & \sigma^2 \tau \\
\hline \hline e & & & & & & \\
\hline \sigma & & & & & & \\
\hline \sigma^2 & & & & & & \\
\hline \tau & & & & & & \\
\hline \sigma \tau & & & & & & \\
\hline \sigma^2 \tau & & & & & & \\
\hline
\end{array}
$$

Key Concepts

Group Theory

Group theory is the area of mathematics that studies algebraic structures known as groups, which consist of a set equipped with a binary operation that satisfies four key properties: closure, associativity, the existence of an identity element, and the existence of inverse elements. This framework is fundamental to various mathematical disciplines and applications, providing insight into the nature of symmetry and structure.

Symmetric Group

The symmetric group is the group of all permutations of a finite set, and it plays a pivotal role in abstract algebra. It exemplifies key group properties and serves as one of the simplest examples of both abelian and non-abelian groups. The symmetric group on three elements, in particular, is a small yet rich structure that illustrates non-commutative behavior and other fundamental group concepts in a manageable setting.

Cayley Table

A Cayley table is a visual representation of the operation of a group, similar to a multiplication table. It systematically lists the results of combining any two elements of the group, making it easier to verify the group’s structural properties such as closure, the existence of identity and inverses, and whether the group operation is commutative or not.

Non-Abelian Group

A non-abelian group is a group in which the operation is not commutative, meaning there exist elements a and b such that a • b is not equal to b • a. This property is crucial in understanding more complex symmetries and operations in mathematics, as many important groups, particularly those arising in fields like physics and geometry, are non-abelian.

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