Symmetric Group on 4 Letters/Cycle Notation
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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 $2$