Subgroup of a symmetric group Sn

In summary, if G is a subgroup of a symmetric group Sn, then either all elements of G are even permutations or exactly half of the elements of G are even permutations. This can be shown by considering the subsets of G consisting of even and odd permutations, and showing that the even subset is a subgroup of G. Additionally, if there exists at least one odd permutation in G, then the even subset acts on the odd subset and the orbit of any element in the odd subset is the entire odd subset.
  • #1
leonamccauley
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Homework Statement


Show that if G is a subgroup of a symmetric group Sn, then either every element of G is an even permutation or else exactly half the elements of G are even permutations.


Homework Equations





The Attempt at a Solution


We have a hint for the problem. If all the elements of G are even, then there is nothing to prove. That is clear because even times even will yield and even number of permutations so there is closure under multiplication.
And if you have even transposition the inverse of it will be in the set as well so clearly if we have an even number of permutations its a subgroup.

If it is not we are to let e, o(1), 0(2),...o(k-1) be the even elements of G..

I am stuck so i looked at S3 a subgroup of S9. S3 has 6 elements.
(1 3 2) (1 2 3) and e are even
(1 2) (1 3) and (2 3) are odd
So exactly half like the theorem states but i am stuck.
 
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  • #2
So, let G be a subgroup of Sn. Let Godd be the subset of G consisting of all odd permutations, and let Geven be the subset of G consisting of all even permutation.

Try to show first that Geven is a subgroup of G (this is somewhat analogous with what you've shown so far).

There are two possibilities:
1) Godd is empty: then G consists only out of even permutation, this is what we wanted to show.

2) There exists an [tex]\sigma\in G_{odd}[/tex]. Now try to show that [tex]G_{odd}=\{\sigma\tau~\vert~\tau \in G_{even}\}[/tex]. Or in more highbrow terminology: Geven acts on Godd, and the orbit of an element in Godd is entire Godd.
 
  • #3
Or maybe just note that if o is an odd member of G and G is a subgroup then oG=G. How does multiplication by o affect the oddness and evenness of members of G?
 

1. What is a subgroup of a symmetric group Sn?

A subgroup of a symmetric group Sn is a subset of the original group that contains only elements that can be obtained by combining two or more elements from the original group. It is a smaller group that still follows the same rules and properties as the original group.

2. How is a subgroup of a symmetric group Sn determined?

A subgroup of a symmetric group Sn is determined by identifying a set of elements that form a smaller group and satisfy the four properties of a subgroup: closure, associativity, identity, and inverse.

3. What is the order of a subgroup in a symmetric group Sn?

The order of a subgroup in a symmetric group Sn is the number of elements in the subgroup. It is always a divisor of the order of the original group and can range from 1 (for the trivial subgroup containing only the identity element) to the order of the original group.

4. Can a subgroup of a symmetric group Sn be isomorphic to the original group?

Yes, a subgroup of a symmetric group Sn can be isomorphic to the original group. This happens when the subgroup contains all the elements of the original group and follows the same group structure and properties.

5. How does the concept of a subgroup of a symmetric group Sn relate to permutation groups?

A subgroup of a symmetric group Sn is a type of permutation group, as it is a subset of the original group that contains only elements that can be obtained by permuting the elements of the original group. The subgroup can be seen as a smaller permutation group within the larger group.

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