Symmetric Group on 4 Letters/Normalizers

Normalizers of the Symmetric Group on 4 Letters
Let $S_4$ denote the Symmetric Group on 4 Letters, whose Cayley table is given as:

Let $\alpha$ denote the permutation in $S_4$ given in cycle notation as $\tuple {1234}$.

The normalizer of $S = \set {\alpha, \alpha^{-1} }$ in $S_4$ is given by:


 * $\map {N_{S_4} } S = \set {e, \alpha, \alpha^2, \alpha^3, \beta, \alpha \beta, \alpha^2 \beta, \alpha^3 \beta}$

where $\beta$ denotes the permutation in $S_4$ given in cycle notation as $\tuple {24}$.