Symmetric Group on 3 Letters/Normal Subgroups
Normal Subgroups of the Symmetric Group on 3 Letters
Let $S_3$ denote the Symmetric Group on 3 Letters, whose Cayley table is given as:
- $\begin{array}{c|cccccc} \circ & e & (123) & (132) & (23) & (13) & (12) \\ \hline e & e & (123) & (132) & (23) & (13) & (12) \\ (123) & (123) & (132) & e & (13) & (12) & (23) \\ (132) & (132) & e & (123) & (12) & (23) & (13) \\ (23) & (23) & (12) & (13) & e & (132) & (123) \\ (13) & (13) & (23) & (12) & (123) & e & (132) \\ (12) & (12) & (13) & (23) & (132) & (123) & e \\ \end{array}$
Consider the subgroups of $S_3$:
The subsets of $S_3$ which form subgroups of $S_3$ are:
\(\displaystyle \) | \(\) | \(\displaystyle S_3\) | |||||||||||
\(\displaystyle \) | \(\) | \(\displaystyle \set e\) | |||||||||||
\(\displaystyle \) | \(\) | \(\displaystyle \set {e, \tuple {123}, \tuple {132} }\) | |||||||||||
\(\displaystyle \) | \(\) | \(\displaystyle \set {e, \tuple {12} }\) | |||||||||||
\(\displaystyle \) | \(\) | \(\displaystyle \set {e, \tuple {13} }\) | |||||||||||
\(\displaystyle \) | \(\) | \(\displaystyle \set {e, \tuple {23} }\) |
Of those, the normal subgroups in $S_3$ are:
- $S_3, \set e, \set {e, \tuple {123}, \tuple {132} }$
Proof
$S_3$ itself is normal in $S_3$ by Group is Normal in Itself.
$\set e$ is normal in $S_3$ by Trivial Subgroup is Normal.
$\set {e, \tuple {12} }$:
\(\displaystyle \tuple {123} \tuple {13} \tuple {123}^{-1}\) | \(=\) | \(\displaystyle \tuple {123} \tuple {13} \tuple {132}\) | |||||||||||
\(\displaystyle \) | \(=\) | \(\displaystyle \tuple {123} \tuple {12}\) | |||||||||||
\(\displaystyle \) | \(=\) | \(\displaystyle \tuple {23}\) | |||||||||||
\(\displaystyle \) | \(\notin\) | \(\displaystyle \set {e, \tuple {12} }\) |
Hence $\set {e, \tuple {12} }$ is not normal in $S_3$.
$\set {e, \tuple {23} }$:
\(\displaystyle \tuple {123} \tuple {23} \tuple {123}^{-1}\) | \(=\) | \(\displaystyle \tuple {123} \tuple {23} \tuple {132}\) | |||||||||||
\(\displaystyle \) | \(=\) | \(\displaystyle \tuple {123} \tuple {13}\) | |||||||||||
\(\displaystyle \) | \(=\) | \(\displaystyle \tuple {12}\) | |||||||||||
\(\displaystyle \) | \(\notin\) | \(\displaystyle \set {e, \tuple {23} }\) |
Hence $\set {e, \tuple {23} }$ is not normal in $S_3$.
$\set {e, \tuple {13} }$:
\(\displaystyle \tuple {123} \tuple {13} \tuple {123}^{-1}\) | \(=\) | \(\displaystyle \tuple {123} \tuple {13} \tuple {132}\) | |||||||||||
\(\displaystyle \) | \(=\) | \(\displaystyle \tuple {123} \tuple {12}\) | |||||||||||
\(\displaystyle \) | \(=\) | \(\displaystyle \tuple {23}\) | |||||||||||
\(\displaystyle \) | \(\notin\) | \(\displaystyle \set {e, \tuple {13} }\) |
Hence $\set {e, \tuple {13} }$ is not normal in $S_3$.
$\set {e, \tuple {123}, \tuple {132} }$:
We have that $\set {e, \tuple {123}, \tuple {132} }$ is the set of even permutations of $S_3$.
Any permutation of the form $\alpha \pi \alpha^{-1}$, for $\pi$ even, is also even.
Thus:
- $\forall \alpha \in S_3: \alpha \pi \alpha^{-1} \in \set {e, \tuple {123}, \tuple {132} }$
Hence $\set {e, \tuple {123}, \tuple {132} }$ is normal in $S_3$.
$\blacksquare$
Sources
- 1965: J.A. Green: Sets and Groups ... (previous) ... (next): $\S 6.6$. Normal subgroups: Example $122$
- 1996: John F. Humphreys: A Course in Group Theory ... (previous) ... (next): Chapter $7$: Normal subgroups and quotient groups: Exercise $2$