Suppose that H is a subgroup of a finite group G and that...

Suppose that $H$ is a subgroup of a finite group $G$ and that $|H|$ and $(|G: H|-1) !$ are relatively prime. Prove that $H$ is normal in $G$. What does this tell you about a subgroup of index 2 in a finite group?

Key Concepts

Normal Subgroup

A normal subgroup is one that is invariant under conjugation by any element of the group. This means that for every element g in the group G and every element h in the subgroup H, the conjugate gHg?¹ equals H. Normal subgroups are fundamental because they allow the construction of quotient groups, which are central to many areas of group theory.

Index of a Subgroup

The index of a subgroup H in a group G, denoted |G : H|, is the number of distinct left (or right) cosets of H in G. This number provides a measure of how large the entire group is relative to the subgroup, and subgroups with small index (for example, index 2) often possess special properties that can simplify many group theoretical arguments.

Group Action by Conjugation

The concept of a group acting on itself by conjugation is a key tool in understanding normality. When G acts on the set of cosets of H, the kernel of this action can be analyzed to determine invariance properties of H. This approach often leads to a homomorphism from G into a symmetric group, and properties of this homomorphism are critical for proofs that conclude normality under certain conditions.

Relatively Prime Condition

The condition that the order of H and the factorial of (|G : H| - 1) are relatively prime is used to restrict the possible prime divisors of |H>. This arithmetic constraint is central in arguments that force certain actions (like conjugation) to be trivial on H, thereby establishing that H is normal in G.

Subgroup of Index 2

A subgroup of index 2 in any finite group is automatically normal due to the fact that there are only two cosets. In this scenario, the nontrivial coset must be the complement of the subgroup, and conjugation necessarily preserves this two-set partition, ensuring that the subgroup remains invariant under conjugation.

Please give Ace some feedback