abuot permutations?

Question

comput the order of each member of A-4??
A-4 is agroup of even permutations
of S-4
plz explain

Answer 1

Here are the elements of A_4.
(123), (132),
(124), (142),
(134), (143),
(234), (243)
(12)(34),
(13)(24)
(14)(23).
and e.
The order of e is, of course, 1 and
(12)(34), (13)(24), (14)(23)
each have order 2, as you can see by squaring each one.
Now look at the pairs in the first 4 rows. In each
row, the square of the first permutation is the second one
and the second one is also the inverse of the first,
as you can see by multiplying them.
So if g is any of these elements
we have g² = g^-1 and g³ = e,
so all such g have order 3.

Answer 2

Let's look at some even permutations and their orders:
a = (12)(34)
a^2 = (12)(34)(12)(34) = id This will always happen when all of the entries in the transpositions of a are different.
b = (12)(23)
b^2 = (12)(23)(12)(23) =(132)
b^3 =(12)(23)(132) = id
We'll do one more:
c = (12)(23)(13)(24)
c^2 = (12)(23)(13)(24)(12)(23)(13)(24) = (234)
c^3 = (12)(23)(13)(24)(234) = id
There are twelve elements in A4. Using what I did above, you should be able to do the remainder. You can send me any questions you have via email.