Symmetric Group on 3 Letters/Normal Subgroups

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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