Calculating $A_{n}$ I'm a beginner in Group theory, and I'm Trying to calculate $A_{4}$, but I couldn't.
I know that this is the subgroup of even permutations in $S_{n}$.
How I can calculate it's all elements?
How I can calculate the all elements of $A_{n}$?
 A: The group $S_4$ consists of permutations with these cyclic structures:

*

*$(1)(1)(1)(1)$- the identity permutation, even.


*$(2)(1)(1)$ - an involution, odd


*$(2)(2)$ - even


*$(3)(1)$ - even


*$(4)$ - odd.
So permutations with cycle structure $(1)(1)(1)(1),(2)(2)$ or $(3)(1)$ form $A_4$.
In general, in $S_n$, for a permutation with cycle structure $(k_1)...(k_p)$ to see if this permutation is even or odd you need to look at the parity of the sum $(k_1-1)+(k_2-1)+...+(k_p-1)$.
A: There are only $12$ elements.  There are $8$ three cycles:  $(123),(132),(124),(142),(234),(243),(134),(143)$.  That leaves three, if we throw in the identity. They are $(12)(34),(13)(24),(14)(23)$.
Recall, an odd cycle is even and an even cycle is odd.  Even means the permutation can be written as a product of an even number of transpositions ($2$-cycles).
A: One concrete way is to consider $S_4$ as the $4\times 4$ permutation matrices (every row/column has exactly one nonzero entry which is equal to one).  Then $A_4$ is the kernel of the determinant map $\det:S_4\to\{\pm1\}$ (i.e. the permutation matrices with determinant 1).
