Prove that the group A4 is not abelian

The Mathematician — Sun, 30 Apr 2023 13:00:07 +0000

Prove that the group

A_4

is not abelian.

Proof. First of all, the alternating group of degree 4 is defined as:

\begin{align*} A_4 = \{1,(123),(124),(132),(134),(142),(143),(234),(243),(12)(34),(13)(24),(14)(23)\}. \end{align*}

To show that

A_4

is not abelian, we need to find a counterexample. That is, we need to find

\sigma, \tau \in A_4

such that

\sigma\tau \neq \tau\sigma

. So, let

\sigma = (123), \tau = (13)(24) \in A_4

. Then

\begin{align*} (123)(13)(24) = 1 \end{align*}

and

\begin{align*} (13)(24)(123) = (142). \end{align*}

But

(123)(13)(24) \neq (13)(24)(123)

, so

A_4

is not abelian.

A_4 Archives - Epsilonify