Question

List left and right cosets of H= {epsilon, alpha} in S3, where epsilon is the identity permutation, alpha is the transposition (2,3). Is H a normal subgroup of S3? Justify.

Answer 1

$$H = \left\{1, \left( \begin{array}{ccc} 1 & 2 & 3 \\ 1 & 3 & 2 \end{array} \right) \right\}$$ $$\left( \begin{array}{ccc} 1 & 2 & 3 \\ 1 & 3 & 2 \end{array} \right) \left( \begin{array}{ccc} 1 & 2 & 3 \\ 1 & 3 & 2 \end{array} \right) = 1$$

Every subgroup where all elements have second order is commutative, and normal. Also, by Second Sylow theorem all Sylow 2-subgroups are Normal.