Symmetric Group is not Abelian

Theorem

Let $S_n$ be the symmetric group of order $n$ where $n \ge 3$.

Then $S_n$ is not abelian.

Let $\alpha \in S_n$ such that $\alpha$ is not the identity mapping.

From Center of Symmetric Group is Trivial, $\alpha$ is not in the center $\map Z {S_n}$ of $S_n$.

Thus $S_n \ne \map Z {S_n}$.

The result follows by definition of abelian group.

$\blacksquare$

Let $a, b, c \in S$.

Let $\alpha$ be the transposition on $S$ which exchanges $a$ and $b$.

Let $\beta$ be the transposition on $S$ which exchanges $b$ and $c$.

Then:

$\alpha \circ \beta$ maps $\tuple {a, b, c}$ to $\tuple {c, a, b}$

while:

$\beta \circ \alpha$ maps $\tuple {a, b, c}$ to $\tuple {b, c, a}$

Thus $\alpha, \beta \in S_n$ such that $\alpha$ does not commute with $\beta$.

Hence the result by definition of abelian group.

$\blacksquare$

1965: J.A. Green: Sets and Groups ... (previous) ... (next): $\S 4.5$. Examples of groups: Example $79$
1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text I$: Algebraic Structures: $\S 7$: Semigroups and Groups: Exercise $7.3$
1967: George McCarty: Topology: An Introduction with Application to Topological Groups ... (previous) ... (next): Chapter $\text{II}$: Groups: The Group Property
1971: Allan Clark: Elements of Abstract Algebra ... (previous) ... (next): Chapter $2$: Examples of Group Structure: $\S 30 \beta$
1974: Robert Gilmore: Lie Groups, Lie Algebras and Some of their Applications ... (previous) ... (next): Chapter $1$: Introductory Concepts: $1$. Basic Building Blocks: $2$. GROUP
1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): Abelian group