Alternating Group on 4 Letters/Subgroups
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Subgroups of the Alternating Group on $4$ Letters
Let $A_4$ denote the alternating group on $4$ letters, whose Cayley table is given as:
- $\begin{array}{c|cccc|cccc|cccc} \circ & e & t & u & v & a & b & c & d & p & q & r & s \\ \hline e & e & t & u & v & a & b & c & d & p & q & r & s \\ t & t & e & v & u & b & a & d & c & q & p & s & r \\ u & u & v & e & t & c & d & a & b & r & s & p & q \\ v & v & u & t & e & d & c & b & a & s & r & q & p \\ \hline a & a & c & d & b & p & r & s & q & e & u & v & t \\ b & b & d & c & a & q & s & r & p & t & v & u & e \\ c & c & a & b & d & r & p & q & s & u & e & t & v \\ d & d & b & a & c & s & q & p & r & v & t & e & u \\ \hline p & p & s & q & r & e & v & t & u & a & d & b & c \\ q & q & r & p & s & t & u & e & v & b & c & a & d \\ r & r & q & s & p & u & t & v & e & c & b & d & a \\ s & s & p & r & q & v & e & u & t & d & a & c & b \\ \end{array}$
The subsets of $A_4$ which form subgroups of $A_4$ are as follows:
Trivial:
\(\ds \) | \(\) | \(\ds \set e\) | Trivial Subgroup is Subgroup | |||||||||||
\(\ds \) | \(\) | \(\ds A_4\) | Group is Subgroup of Itself |
\(\ds \) | \(\) | \(\ds \set {e, t}\) | as $t^2 = e$ | |||||||||||
\(\ds \) | \(\) | \(\ds \set {e, u}\) | as $u^2 = e$ | |||||||||||
\(\ds \) | \(\) | \(\ds \set {e, v}\) | as $v^2 = e$ |
\(\ds \) | \(\) | \(\ds \set {e, a, p}\) | as $a^2 = p$, $a^3 = a p = e$ | |||||||||||
\(\ds \) | \(\) | \(\ds \set {e, b, s}\) | as $b^2 = s$, $b^3 = b s = e$ | |||||||||||
\(\ds \) | \(\) | \(\ds \set {e, c, q}\) | as $c^2 = q$, $c^3 = c q = e$ | |||||||||||
\(\ds \) | \(\) | \(\ds \set {e, d, r}\) | as $d^2 = r$, $d^3 = d r = e$ |
\(\ds \) | \(\) | \(\ds \set {e, t, u, v}\) | Klein $4$-Group |
Examples of Subgroups
Order $3$
Let $P$ denote the subset of $A_4$:
- $P := \set {e, a, p}$
Then $P$ is a subgroup of $A_4$.
Its left cosets are:
\(\ds P\) | \(=\) | \(\ds \set {e, a, p}\) | ||||||||||||
\(\ds t P\) | \(=\) | \(\ds \set {t, b, q}\) | ||||||||||||
\(\ds u P\) | \(=\) | \(\ds \set {u, c, r}\) | ||||||||||||
\(\ds v P\) | \(=\) | \(\ds \set {v, d, s}\) |
Its right cosets are:
\(\ds P\) | \(=\) | \(\ds \set {e, a, p}\) | ||||||||||||
\(\ds P t\) | \(=\) | \(\ds \set {t, c, s}\) | ||||||||||||
\(\ds P u\) | \(=\) | \(\ds \set {u, d, q}\) | ||||||||||||
\(\ds P v\) | \(=\) | \(\ds \set {v, b, r}\) |
Sources
- 1996: John F. Humphreys: A Course in Group Theory ... (previous) ... (next): Chapter $9$: Permutations: Exercise $3$