Orbits of a normal subgroup of a finite group

In summary, the homework statement states that if G is a finite group which acts transitively on X, and H is a normal subgroup of G, then the orbits of the induced action of H on X all have the same size.
  • #1
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Homework Statement


If G is a finite group which acts transitively on X, and if H is a normal subgroup of G, show that the orbits of the induced action of H on X all have the same size.

The Attempt at a Solution


By the Orbit-Stabilizer theorem the size of the orbit induced by H on X is a divisor of H. This could certainly help... And H is normal, therefore H is the stabilizer of the action of conjugation. Plus the fact that points in the same orbit have conjugate stabilizers... I don't know how to put the elements together... Can anyone hint me?
 
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  • #2
You really don't need anything fancy here. If x is an element of X then the orbit under H is Hx. If y is another element of X then there is a g in G such that y=gx (since G is transitive). So the orbit of y, Hy=Hgx. Now use that H is normal.
 
  • #3
I think I see it now... Hy=Hgx=gHg-1gx=gHx, a left coset of Hx, which has the same size as Hx, since each product of g with a member of Hx is unique and together all these products fill out gHx. The number of products is exactly as big as the size of Hx. Therefore gHx and Hx have the same size.

Correct? Thanks (again!) for your help!
 
  • #4
3029298 said:
I think I see it now... Hy=Hgx=gHg-1gx=gHx, a left coset of Hx, which has the same size as Hx, since each product of g with a member of Hx is unique and together all these products fill out gHx. The number of products is exactly as big as the size of Hx. Therefore gHx and Hx have the same size.

Correct? Thanks (again!) for your help!

Sure. That's correct. Not so hard, was it?
 
  • #5
no :)
 

1. What is a normal subgroup of a finite group?

A normal subgroup of a finite group is a subgroup that is invariant under conjugation by any element of the group. In other words, if we take any element from the group and conjugate it with an element from the normal subgroup, the result will always be an element of the normal subgroup.

2. How do you determine if a subgroup is normal?

To determine if a subgroup is normal, we can use the normal subgroup test. This test states that a subgroup H of a group G is normal if and only if g-1hg is an element of H for all g in G and h in H. In other words, if the conjugate of any element in H by any element in G is still in H, then H is a normal subgroup of G.

3. How do orbits relate to normal subgroups?

In the context of group theory, the orbit of an element g in a group G under a subgroup H is the set of all elements in G that can be obtained by conjugating g with elements from H. Normal subgroups are important because they are precisely the subgroups that partition the group into orbits. In other words, elements in the same orbit under a normal subgroup are conjugate to each other, and elements in different orbits are not conjugate.

4. Can a finite group have more than one normal subgroup?

Yes, a finite group can have multiple normal subgroups. In fact, a group always has at least two normal subgroups: the trivial subgroup (containing only the identity element) and the entire group itself. Other examples of normal subgroups include the center of a group, which consists of all elements that commute with every other element in the group, and the commutator subgroup, which consists of all elements that can be written as the commutator of two elements in the group.

5. Are normal subgroups important in applications of group theory?

Yes, normal subgroups are very important in applications of group theory. They are used to classify and study finite groups, and they also play a crucial role in the theory of group actions, which has applications in areas such as geometry, physics, and cryptography. Additionally, normal subgroups are essential in the study of quotient groups, which are important in the construction of new groups from existing ones.

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