Isomorphism Theorems and Quotient Groups

The true power of group theory lies not just in cataloging groups, but in understanding how they relate to one another through quotient structures and homomorphisms. This lesson explores the internal anatomy of groups, focusing on the conditions under which we can “divide” a group to simplify its structure.

Normal Subgroups and Quotient Groups

For any subgroup $H\le G$, we can define the left cosets $gH=\{gh\mid h\in H\}$ and right cosets $Hg=\{hg\mid h\in H\}$.

Lagrange’s Theorem

If $G$ is a finite group and $H$ is a subgroup of $G$, then the order of $H$ divides the order of $G$: $|G|=|H|\cdot [G:H]$ where $[G:H]$ is the number of distinct cosets (the index of $H$ in $G$).

Normal Subgroups

A subgroup $N\le G$ is called a normal subgroup (denoted $N⊴G$) if its left and right cosets coincide for all $g\in G$: $gN=Ng\forall g\in G$ Equivalent condition: $gN{g}^{-1}\subseteq N$ for all $g\in G$.

The Quotient Group $G/N$

If $N⊴G$, the set of cosets $G/N=\{gN\mid g\in G\}$ forms a group under the operation $(aN)(bN)=(ab)N$. This is the “quotient group” or “factor group,” where $N$ acts as the identity element.

The First Isomorphism Theorem

This is the most fundamental result in algebraic structural analysis. It relates homomorphisms to quotient groups.

Theorem: Let $\varphi :G\to H$ be a group homomorphism. Then the kernel of $\varphi$ is normal in $G$, and the image of $\varphi$ is isomorphic to the quotient of $G$ by the kernel: $G/\ker (\varphi )\cong \text{Im}(\varphi )$

This theorem tells us that every homomorphism can be decomposed into a natural projection onto a quotient group followed by an embedding.

The Second (Diamond) Isomorphism Theorem

Let $G$ be a group, $H\le G$ a subgroup, and $N⊴G$ a normal subgroup. Then:

1. $HN=\{hn\mid h\in H,n\in N\}$ is a subgroup of $G$.
2. $H\cap N$ is a normal subgroup of $H$.
3. $H/(H\cap N)\cong HN/N$.

The “diamond” name comes from the lattice diagram of these subgroups, where $HN$ is at the top, $N$ and $H$ are in the middle, and $H\cap N$ is at the bottom.

The Third (Freshman) Isomorphism Theorem

Let $G$ be a group, and let $H$ and $K$ be normal subgroups of $G$ such that $K\subseteq H\subseteq G$. Then:

1. $H/K$ is a normal subgroup of $G/K$.
2. $(G/K)/(H/K)\cong G/H$.

This theorem allows us to simplify “quotients of quotients.”

The Lattice (Correspondence) Theorem

Let $N⊴G$. There is a one-to-one correspondence between the subgroups of $G$ containing $N$ and the subgroups of the quotient group $G/N$. Furthermore, this correspondence preserves normality, indices, and intersections. This means that the structure of $G/N$ reflects a specific “slice” of the structure of $G$.

Simple Groups and Composition Series

A group is simple if it has no proper non-trivial normal subgroups. Simple groups are the “atoms” of group theory.

• The cyclic groups $\mathbb{Z}/p\mathbb{Z}$ for prime $p$ are simple.
• The alternating group $A_n$ for $n\ge 5$ is simple.

The Jordan-Hölder Theorem states that any finite group can be broken down into a sequence of simple groups (a composition series), and these simple groups are unique up to reordering. This led to the monumental “Classification of Finite Simple Groups,” completed in the early 21st century.

Group Action and Orbits

Advanced group theory often involves groups “acting” on sets. A group action of $G$ on a set $X$ is a map $G\times X\to X$ such that $e\cdot x=x$ and $(gh)\cdot x=g\cdot (h\cdot x)$.

1. Orbit: $G\cdot x=\{g\cdot x\mid g\in G\}$.
2. Stabilizer: $G_x=\{g\in G\mid g\cdot x=x\}$.
3. Orbit-Stabilizer Theorem: $|G\cdot x|=[G:G_x]$.

Python: Exploring Cosets

import itertools

def get_cosets(G, H):
    \"\"\"
    G is a list of elements, H is a list of elements (subgroup).
    This computes all left cosets gH.
    \"\"\"
    cosets = []
    seen = set()
    for g in G:
        if g not in seen:
            coset = sorted([(g + h) % len(G) for h in H]) # Example with Z_n
            cosets.append(coset)
            for x in coset:
                seen.add(x)
    return cosets

# Z_12 with subgroup generated by 3: {0, 3, 6, 9}
G_elements = list(range(12))
H_subgroup = [0, 3, 6, 9]
print(f"Cosets: {get_cosets(G_elements, H_subgroup)}")

Exercise