Symmetric Group on 4 Letters/Cycle Notation

Cycle Notation for Symmetric Group on $4$ Letters

The alternating group on $4$ letters can be given in cycle notation, and its elements assigned arbitrary labels from $1$ to $23$ as follows:

\(\ds e\)

\(:=\)

\(\ds \text { the identity mapping}\)

\(\ds t_{12}\)

\(:=\)

\(\ds \tuple {1 2}\)

\(\ds t_{23}\)

\(:=\)

\(\ds \tuple {2 3}\)

\(\ds r_{132}\)

\(:=\)

\(\ds \tuple {1 3 2}\)

\(\ds r_{123}\)

\(:=\)

\(\ds \tuple {1 2 3}\)

\(\ds t_{13}\)

\(:=\)

\(\ds \tuple {1 3}\)

\(\ds t_{34}\)

\(:=\)

\(\ds \tuple {3 4}\)

\(\ds v_a\)

\(:=\)

\(\ds \tuple {1 2} \tuple {3 4}\)

\(\ds r_{243}\)

\(:=\)

\(\ds \tuple {2 4 3}\)

\(\ds f_{1432}\)

\(:=\)

\(\ds \tuple {1 4 3 2}\)

\(\ds f_{1243}\)

\(:=\)

\(\ds \tuple {1 2 4 3}\)

\(\ds r_{143}\)

\(:=\)

\(\ds \tuple {1 4 3}\)

\(\ds r_{234}\)

\(:=\)

\(\ds \tuple {2 3 4}\)

\(\ds f_{1342}\)

\(:=\)

\(\ds \tuple {1 3 4 2}\)

\(\ds t_{24}\)

\(:=\)

\(\ds \tuple {2 4}\)

\(\ds r_{142}\)

\(:=\)

\(\ds \tuple {1 4 2}\)

\(\ds v_b\)

\(:=\)

\(\ds \tuple {1 3} \tuple {2 4}\)

\(\ds f_{1423}\)

\(:=\)

\(\ds \tuple {1 4 2 3}\)

\(\ds f_{1234}\)

\(:=\)

\(\ds \tuple {1 2 3 4}\)

\(\ds r_{134}\)

\(:=\)

\(\ds \tuple {1 3 4}\)

\(\ds r_{124}\)

\(:=\)

\(\ds \tuple {1 2 4}\)

\(\ds t_{14}\)

\(:=\)

\(\ds \tuple {1 4}\)

\(\ds f_{1324}\)

\(:=\)

\(\ds \tuple {1 3 2 4}\)

\(\ds v_c\)

\(:=\)

\(\ds \tuple {1 4} \tuple {2 3}\)

Sources

• 1996: John F. Humphreys: A Course in Group Theory ... (previous) ... (next): Chapter $9$: Permutations: Exercise $2$