1. ## Permutations (Want to check my answer)

I just want to make sure my answer is correct

Questions: List all even permutations of symmetry group S4

Answer: There are 12 even permutations in S4
(1234)(1342)(1423)(2143)(3124)(4132)(2431)(2314)(4 213)(3241)(3412)(4321)

2. Originally Posted by mathlg
I just want to make sure my answer is correct

Questions: List all even permutations of symmetry group S4

Answer: There are 12 even permutations in S4
(1234)(1342)(1423)(2143)(3124)(4132)(2431)(2314)(4 213)(3241)(3412)(4321)

Rememeber that the cardinality of the finite 'alternating group" is $\displaystyle \frac{n!}{2}$.
In this case $\displaystyle n=4$ thus,
$\displaystyle \frac{4!}{2}=\frac{24}{2}=12$
Thus there are 12 even premutations.

For the second part, all the premutations you named are odd!
Because,
$\displaystyle (a,b,c,d)=(a,b)(b,c)(c,d)$
The number of transpositions here is an odd number!
Also, the premutation
$\displaystyle (a,b,c,d)=(d,a,b,c)$

Thus, there are two mistakes, first they are all odd

and not all are distinct-which basically means you did not list all the odd premutations.

3. A4 = {(1,2,3), (1,3,2), (1,2,4), (1,4,2), (1,3,4), (1,4,3), (2,3,4), (2,4,3), (1,2)(3,4), (1,3)(2,4), (1,4)(2,3),e}.