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Abstract Algebra

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Left Cosets of $S_3$

ABSALG-Q53YVJ

Let $S_3$ be the symmetric group on 3 letters.

Using the convention of multiplying cycles from right to left, determine the left cosets of the subgroup $H=\{(1), (23)\}$ in $S_3$.

For example, $(12)(23)=(123)$.

A

$H$, $\{(12), (123)\}$, $\{(13), (132)\}$

B

$H$, $\{(12), (132)\}$, $\{(13), (123)\}$

C

$H$, $\{(12), (13)\}$, $\{(123), (132)\}$

D

$\{(1), (13)\}$, $\{(12), (132)\}$, $\{(23), (123)\}$