symmetric group Archives - Epsilonify Best solutions on internet! Sat, 16 Sep 2023 22:42:36 +0000 en-US hourly 1 https://wordpress.org/?v=6.4.5 https://www.epsilonify.com/wp-content/uploads/2022/09/cropped-E-M7-32x32.png symmetric group Archives - Epsilonify 32 32 Prove that the symmetric group of degree at least 3 is non-abelian https://www.epsilonify.com/mathematics/group-theory/the-symmetric-group-of-degree-at-least-3-is-non-abelian/ https://www.epsilonify.com/mathematics/group-theory/the-symmetric-group-of-degree-at-least-3-is-non-abelian/#respond Tue, 18 Apr 2023 13:00:41 +0000 https://www.epsilonify.com/?p=2180 The symmetric group is non-abelian for . Proof. Since we need to look at degrees at least , we can take . Now note that: and In order to be abelian, it must be that , but that is not the case since Therefore, the symmetric group is non-abelian for .

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]]> The symmetric group S_n is non-abelian for n \geq 3.

Proof. Since we need to look at degrees at least 3, we can take (12),(13) \in S_n. Now note that: 
\begin{align*}
(12)(13) = (132)
\end{align*}
and 
\begin{align*}
(13)(12) = (123).
\end{align*}
In order to be abelian, it must be that (12)(13) = (13)(12), but that is not the case since 
\begin{align*}
(12)(13) = (132) \neq (123) = (13)(12).
\end{align*}
Therefore, the symmetric group S_n is non-abelian for n \geq 3.

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