Proof on Normal Subgroups and Cosets in Group Theory

In summary, the problem is to prove that if a subgroup H of a permutation group has an intersection A with the alternating group of permutation n, and A is not equal to H, then A is a normal subgroup of H with index two in H. The hint is to consider an element g in H but not in A, and show that gA and A are two cosets of A in H. It is also important to note that the alternating group only consists of even permutations, so if A is not equal to H, then H must contain an odd permutation. The key is to show that half of the elements in H are even permutations and half are odd, and the hint given by the professor is to use the fact that g
  • #1
frinny913
4
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This is a proof I am struggling on ...

Let H be a subgroup of the permutation of n and let A equal the intersection of H and the alternating group of permutation n. Prove that if A is not equal to H, than A is a normal subgroup of H having index two in H.

My professor gave me the hint to begin by letting g be in H but not in A and then showing that gA and A are two cosets of A in H.
 
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  • #2
The alternating group consists of all even permutations, right? If A is not equal to H, then H must contain an odd permutation, g, right? What you want to show is that half the elements of H are even permutations and half are odd. Hint: gH=H. That makes A a subgroup of H of index 2.
 

1. What is a normal subgroup?

A normal subgroup is a subgroup of a group that is invariant under conjugation by elements of the larger group. In other words, if an element of the larger group is conjugated by an element of the normal subgroup, the resulting element is still in the normal subgroup.

2. How do you prove that a subgroup is normal?

To prove that a subgroup is normal, you must show that for any element in the larger group, its conjugate by any element in the subgroup is also in the subgroup. This can be done by directly showing that the subgroup is invariant under conjugation, or by using the fact that normal subgroups are the kernels of group homomorphisms.

3. What are cosets in group theory?

Cosets are subsets of a group formed by multiplying a fixed subgroup by all the elements of the larger group. In other words, for a subgroup H of a group G, the coset gH is the set of all elements obtained by multiplying every element of H by g.

4. How do cosets relate to normal subgroups?

In a normal subgroup, the left and right cosets are the same. In other words, for a normal subgroup H of a group G, the left coset gH is the same as the right coset Hg for any element g in G. This is not necessarily true for non-normal subgroups.

5. How is the concept of normal subgroups and cosets used in group theory?

The concepts of normal subgroups and cosets are used to understand the structure of groups and to prove theorems about groups. They are also used in applications such as cryptography and symmetry. In particular, the quotient group (or factor group) is formed by the cosets of a normal subgroup, and it has many important properties that can help solve problems in group theory.

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