Question

Obtain the left cosets of V₄= {e, (1 2) (3 4), (1 3) (2 4), (1 4) (2 3)}in A₄.

Answer 1

There are $|A_4/V_4|=\frac{|A_4|}{|V_4|}=\frac{12}{4}=3$ different left cosets of the Klein four-subgroup $V_4= \{e, (1 2) (3 4), (1 3) (2 4), (1 4) (2 3)\}$ in the alternating group $A_4.$ Let us find them:

$eV_4=V_4=\{e, (1 2) (3 4), (1 3) (2 4), (1 4) (2 3)\},$

$(123)V_4=(123)\{e, (1 2) (3 4), (1 3) (2 4), (1 4) (2 3)\}=\{(123),(134),(243),(142)\},$ and

$(132)V_4=(132)\{e, (1 2) (3 4), (1 3) (2 4), (1 4) (2 3)\}=\{(132),(234),(124),(143)\}.$