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Gear300
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Changed to a different question:
Can anyone provide a proof for the order of alternating groups |An| = ½(n!)?
Can anyone provide a proof for the order of alternating groups |An| = ½(n!)?
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Gear300 said:Changed to a different question:
Can anyone provide a proof for the order of alternating groups |An| = ½(n!)?
sutupidmath said:[tex]\beta_1\beta_j, j=1,2,3,...,k[/tex] (there is some extra work here to show that all these elements are indeed unique, but it is not difficult to establish it. a proof by contradiction would work)
sutupidmath said:The way i like to show this goes something along these lines:
THe whole idea is to show that there are as many odd permutations as there are even permutation in Sn. So, let[tex]S_n= ( \alpha_i,\beta_j)[/tex] which represents the set of even and odd permutations.
where [tex]\alpha_i, i=1,2,3,...,r[/tex] and [tex]\beta_j,j=1,2,3,...,k[/tex] are even and odd permutations respectively.
Now, let's consider the following:[tex]\beta_1\beta_j, j=1,2,3,...,k[/tex] (there is some extra work here to show that all these elements are indeed unique, but it is not difficult to establish it. a proof by contradiction would work) ---------So, we know that the multiplication of odd permutations is an even permutation, so we know that in our set Sn, we have the following relation:matt grime said:...multiplication by a group element is a bijection is almost the definition of a group.
[tex]|k|\leq|r|------(1)[/tex]
Now, consider the following:[tex]\beta_1\alpha_i,i=1,2,...,r[/tex]
so all these permutations now are odd. From this we get the following relation:
[tex]|r|\leq |k|-----(2)[/tex]From (1) &(2) we get the following:[tex]|k|\leq |r| \leq |k|=>|k|=|r|[/tex]
Which means that the number of even and odd permutations in Sn is equal. ---------Now, since |Sn|=n! => |An|=n!/2matt grime said:...there exists an injection from the set of odd elements to even elements, and vice versa, hence they have the same cardinality...
An alternating group, denoted as An, is a type of permutation group that consists of even permutations. These are permutations that can be decomposed into an even number of transpositions (swaps of two elements).
An alternating group of order n, denoted as |An|, has n! elements. This is half the number of elements in the symmetric group of order n, denoted as |Sn|, which has n! elements.
The proof for the order of an alternating group is based on the concept of cosets. A coset is a subset of a group that is obtained by multiplying all elements of a subgroup by a fixed element. By showing that the alternating group of order n has exactly n!/2 distinct cosets, it can be proved that its order is 1/2(n!), as each coset contains exactly 2 elements.
The order of an alternating group is important in various areas of mathematics and science. It is used in the study of group theory, which has applications in fields such as chemistry, physics, and computer science. The alternating group also has connections to other mathematical concepts, such as symmetric polynomials and the representation theory of finite groups.
Yes, an example of an alternating group of order n is the alternating group of order 4, denoted as A4. It consists of the even permutations of the set {1, 2, 3, 4}, which are:
(1 2)(3 4), (1 3)(2 4), (1 4)(2 3), (1 2 3), (1 3 2), (1 2 4), (1 4 2), (1 3 4), (1 4 3), (2 3 4), (2 4 3).