Subgroups and normal subgroups

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In summary, using Lagrange's Theorem and the fact that G acts on the set of left cosets of N by conjugation, it can be proven that H is a subgroup of N. Additionally, the first isomorphism theorem may also be utilized. Studying the group G/N and N can provide valuable insights about H.
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Let G be a group, let N be normal in G and let H be a subgroup of G. Assume that G/N and H are finite and that gcd(|G/N|,|H|)=1. Prove that H is a subgroup of N.

I was thinking about using Lagrange Theorem. and maybe using the fact that G may act on the set of left cosets (G/N) by conjugation.
and the find the kernel of that action and then maybe use the first isomorphism theorem.

But I don't get very far with that.
 
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H also acts on G/N by left multiplication. Furthermore, every element of H is in one of the the left cosets of N.
 
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Maybe looking at G isn't the right way to go. What can you learn about H by studying the group G/N? What about N?
 

1. What is a subgroup and how is it different from a normal subgroup?

A subgroup is a subset of a group that satisfies the same group axioms as the larger group. A normal subgroup is a subgroup that is invariant under conjugation, meaning that for any element in the larger group, its conjugate also belongs to the normal subgroup. In other words, a normal subgroup is a subgroup that is "symmetric" within the larger group.

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

To determine if a subgroup is normal, you can use the normal subgroup test. This states that a subgroup H of a group G is normal if and only if for all elements g in G and h in H, ghg-1 is also in H. In simpler terms, if the subgroup is closed under conjugation, it is a normal subgroup.

3. What is the significance of normal subgroups?

Normal subgroups have many important properties in group theory. They are the building blocks for quotient groups, which are formed by dividing out a normal subgroup from a larger group. They also play a crucial role in the classification of finite simple groups.

4. Can a subgroup be both normal and not normal at the same time?

No, a subgroup can either be normal or not normal, but not both at the same time. A subgroup cannot satisfy the conditions for normality (being invariant under conjugation) and not satisfy them at the same time.

5. Are all subgroups normal subgroups?

No, not all subgroups are normal subgroups. A normal subgroup is a special type of subgroup that has additional properties, such as being invariant under conjugation. Subgroups that do not satisfy these properties are not considered normal subgroups.

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