Alternating Group is Simple except on 4 Letters/Lemma 3

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Theorem

Let $n$ be an integer such that $n \ge 5$.

Let $A_n$ denote the alternating group on $n$ letters.

Let $\rho \in S_n$ be an arbitrary $3$-cycle.

Let $\N_n$ denote the initial segment of the natural numbers $\set {0, 1, \ldots, n - 1}$.

Let $i, j, k \in \N_n$ be such that $\rho = \tuple {i, j, k}$.

Then there exists an even permutation $\sigma \in A_n$ such that $\map \sigma 1 = i$, $\map \sigma 2 = j$ and $\map \sigma 3 = k$.

Proof

We will proceed by cases.

We have that $\card {\set {1, 2, 3} \cap \set {i, j, k} }$ is either $0$, $1$, $2$ or $3$.

Case $1$

$\card {\set {1, 2, 3} \cap \set {i, j, k} } = 0$ (this case is only possible when $n \ge 6$).

The permutation $\sigma = \tuple {1, i, 2, j} \tuple {3, k}$ is even (by Parity of K-Cycle and Sign of Composition of Permutations) and has the desired property.

Case $2$

$\card {\set {1, 2, 3} \cap \set {i, j, k} } = 1$.

Without loss of generality, suppose that $\set {1, 2, 3} \cap \set {i, j, k} = \set 1$.

We have two further sub-cases: $i = 1$ or $i \ne 1$.

If $i = 1$, then, the permutation $\sigma = \tuple {2, j} \tuple {3, k}$ is even (by Parity of K-Cycle and Sign of Composition of Permutations) and has the desired property.

Now suppose $i \ne 1$.

Without loss of generality, assume that $j = 1$.

Then the permutation $\sigma = \tuple {1, i, 3, k, 2}$ is even (by Parity of K-Cycle) and has the desired property.

Case $3$

$\card {\set {1, 2, 3} \cap \set {i, j, k} } = 2$.

Without loss of generality, assume that $\set {1, 2, 3} \cap \set {i, j, k} = \set {1, 2}$.

We have three further sub-cases::

$i = 1 \land j = 2$

$\paren {i = 1 \land j \ne 2} \lor \paren {i \ne 1 \land j = 2}$

or:

$i \ne 1 \land j \ne 2$.

(In other words, if $\sigma$ exists, it either fixes $2$, $1$, or $0$ letters of those in $\set {1, 2, 3}$.)

Suppose $i = 1 \land j = 2$.

Since $n \ge 5$, there exists an element $l \in \N_n \setminus \set {1, 2, 3, k}$.

Then, the permutation $\sigma = \tuple {3, k, l}$ is even (by Parity of K-Cycle) and has the desired property.

Now suppose $\paren {i = 1 \land j \ne 2} \lor \paren {i \ne 1 \land j = 2}$.

Without loss of generality, assume $\paren {i = 1 \land j \ne 2}$.

This implies $k = 2$.

Then, the permutation $\sigma = \tuple {2, j, 3}$ is even (by Parity of K-Cycle) and has the desired property.

Now suppose $i \ne 1 \land j \ne 2$.

If $i = 2$ and $j = 1$, then the permutation $\sigma = \tuple {1, 2} \tuple {3, k}$ is even (by Parity of K-Cycle and Sign of Composition of Permutations) and has the desired property.

Otherwise, without loss of generality, assume $i = 2$ and $k = 1$.

Since $n \ge 5$, there exists an element $l \in \N_n \setminus \set {1, 2, 3, k}$.

Then, the permutation $\sigma = \tuple {1, 2, j, l, 3}$ is even and has the desired property.

Case $4$

$\card {\set {1, 2, 3} \cap \set {i, j, k} } = 3$.

That is:

$\set {i, j, k} = \set {1, 2, 3}$

We have that a (not necessarily even) permutation $\sigma$ has the desired property if and only if it can be written as a disjoint product $\sigma = \beta\alpha$, where $\alpha$ is the permutation:

$ \begin{pmatrix} 1 & 2 & 3 & 4 & \cdots & n \\ i & j & k & 4 & \cdots & n \\ \end{pmatrix}$

If $\alpha$ fixes all letters, $\alpha$ is the identity.

Take $\beta$ to be the identity.

Then $\sigma$ will be the identity permutation, so by Identity Mapping on Symmetric Group is Even Permutation, it will be even.

If $\alpha$ fixes one letter, $\alpha$ will be a transposition.

Since $n \ge 5$, we can take $\beta$ to be another transposition disjoint from $\alpha$.

Then $\sigma$ will be the product of two disjoint transpositions.

Therefore, by Parity of K-Cycle and Sign of Composition of Permutations, $\sigma$ will be even.

If $\alpha$ fixes no letters, $\alpha$ will be a $3$-cycle.

Take $\beta$ to be the identity permutation.

Then $\sigma = \alpha$.

By Parity of K-Cycle, $\sigma$ will be even.

In all cases, we found an even permutation $\sigma$ with the desired property.

Also, $\rho$ was arbitrary.

Hence a $\sigma$ as described above always exists for any $3$-cycle.

$\blacksquare$