# Subgroups and normal subgroups

• math8
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.
math8
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.

H also acts on G/N by left multiplication. Furthermore, every element of H is in one of the the left cosets of N.

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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