Let $K=\{ (1), (12)(34),(13)(24),(14)(23)\}$ and let $G=S_4$. Determine whether the following classes are disjoint or identical

Question

Let $K=\{ (1), (12)(34),(13)(24),(14)(23)\}$ and let $G=S_4$. $K$ is a subgroup of $S_4$.

Determine whether the following are disjoint or identical:

a) $K(12)$ and $K(34)$

b) $K(1234)$ and $K(1324)$

---

for $a)$

not fond of cyclic multiplication would rather due them in matrices but Ill try

For right coset $K(12)$

$$\begin{aligned} (1)*(12) &=(12) && {\text{ok}} \\ (12)(34) *(12)&=(1)(2)(34) &&{\text{ok}} \\ (13)(24)* (12)&=(1423) &&{\text{ok}} \\ (14)(23)*(12)&=(1324) &&{\text{better now}} \end{aligned} $$ For right coset $K(34)$

$$\begin{aligned} (1)*(34) &=(34) &&{\text{ok}} \\ (12)(34) *(34)&=(12)(3)(4) &&{\text{ok}} \\ (13)(24)* (34)&=(1324) &&{\text{ok}} \\ (14)(23)*(34)&=(1423) \end{aligned} $$

They look identical might have made a mistake since they don't completely match up.

I need to verify that for $b$ it is disjoint.

Answer 1

Maybe look at it more theorethically. We have $Kx = Ky \iff Kxy^{-1}=K \iff xy^{-1} \in K)$. So for a) we have $x = (1,2)$ and $y = y^{-1}=(3,4)$ so $xy^{-1}=(1,2)(3,4)$, but $(1,2)(3,4) \in K$, by definition of $K$, so the cosets are equal. For b) we have $xy^{-1} = (1,2,3,4)(1,3,2,4)^{-1} = (1,3,2) \notin K$, so the cosets are different.