Proving H is Normal in Finite Group with One Subgroup

Thread starter joecoz88
Start date Oct 31, 2008
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In summary, to prove that H is normal in a finite group with one subgroup, one can use the definition of a normal subgroup and show that the left and right cosets of H are equal for every element in the group. This is important because it helps us understand the structure of the group and has various applications in mathematics. Some properties of normal subgroups include being normal in itself, forming a quotient group, and being the kernel of a group homomorphism. Additionally, H can still be a normal subgroup in a non-finite group with one subgroup as long as it satisfies the criteria of being invariant under conjugation.

Oct 31, 2008

joecoz88

Homework Statement

If G is a finite group that has exactly one subgroup H of a given order, then H is normal.

Homework Equations

N/A

The Attempt at a Solution

I cannot figure out what makes a subgroup H special if it is the only one of a given order...

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Oct 31, 2008

e(ho0n3

Pick a g in G. What is the order of g^-1Hg?

1. How do you prove that H is normal in a finite group with one subgroup?

To prove that H is normal in a finite group with one subgroup, you can use the definition of a normal subgroup. This means showing that for every element in the finite group, the left and right cosets of H are the same. You can also use the fact that conjugate subgroups are equal in a normal subgroup.

2. What is the definition of a normal subgroup?

A subgroup H of a group G is considered normal if for any element g in G, the left and right cosets of H are the same. In other words, for all g in G, gH = Hg. This means that H is invariant under conjugation by elements in G.

3. Why is proving H is normal important in a finite group with one subgroup?

Proving that H is normal in a finite group with one subgroup is important because it allows us to understand the structure of the group better. It also helps us to identify important properties of the group and make computations easier. Moreover, the concept of normal subgroups is crucial in the study of group theory and has many applications in different areas of mathematics.

4. What are some properties of normal subgroups?

Some properties of normal subgroups include:

A group G is normal in itself.
If H is normal in G, then the quotient group G/H is a group.
If G is abelian, then every subgroup of G is normal.
If N is a subgroup of G, then N is normal in G if and only if N is the kernel of a group homomorphism from G to another group.

5. Can H be a normal subgroup in a non-finite group with one subgroup?

Yes, H can still be a normal subgroup in a non-finite group with one subgroup. The definition of a normal subgroup does not depend on the finiteness of the group. As long as H satisfies the criteria of being invariant under conjugation, it can be considered a normal subgroup in any group, finite or non-finite.