Alternating Group is Normal Subgroup of Symmetric Group
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Theorem
Let $n \ge 2$ be a natural number.
Let $S_n$ denote the symmetric group on $n$ letters.
Let $A_n$ be the alternating group on $n$ letters.
Then $A_n$ is a normal subgroup of $S_n$ whose index is $2$.
Proof
We have that $\map \sgn {S_n}$ is onto $C_2$.
Thus from the First Isomorphism Theorem, $A_n$ consists of the set of even permutations of $S_n$.
The result follows from Subgroup of Index 2 is Normal.
$\blacksquare$
Examples
When $n = 2$, we have that:
- $S_2 \cong C_2$
and:
- $A_n = \set {e_{S_2} }$
Sources
- 1965: J.A. Green: Sets and Groups ... (previous) ... (next): $\S 7.4$. Kernel and image: Example $142$
- 1971: Allan Clark: Elements of Abstract Algebra ... (previous) ... (next): Chapter $2$: The Symmetric Groups: $\S 81$
- 1996: John F. Humphreys: A Course in Group Theory ... (previous) ... (next): Chapter $9$: Permutations: Definition $9.19$: Remark
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): alternating group
- 2002: John B. Fraleigh: A First Course in Abstract Algebra (7th ed.): Chapter $13$: $32 \, \text{a}$
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): alternating group