Factor Group

Definition

Let $G$ be a group and let $N$ be a normal subgroup of $G$, written $N \trianglelefteq G$. The factor group of $G$ by $N$ is the set of all left cosets of $N$ in $G$:

$$G/N = \{gN \mid g \in G\}$$

with multiplication defined by

$$(gN)(hN) = (gh)N$$

for all $g,h \in G$.

This operation is well-defined only when $N$ is normal in $G$. The factor group $G/N$ is itself a group under this operation.

A factor group can be thought of as the result of “collapsing” every element of $N$ to the identity and treating all elements in the same coset as one single object.

Main Content

1. Cosets and Their Role in Building a Factor Group

• A coset is formed by multiplying every element of a subgroup by a fixed element of the group. If $N$ is a subgroup of $G$ and $g \in G$, then the left coset is $$gN = \{gn \mid n \in N\}.$$ Cosets partition the group into disjoint, equally sized subsets.

• In the factor group, each coset becomes one element of the new group. For example, if $G = \mathbb{Z}$ and $N = 4\mathbb{Z}$, then the cosets are: $$0+4\mathbb{Z},\ 1+4\mathbb{Z},\ 2+4\mathbb{Z},\ 3+4\mathbb{Z}.$$ These are the elements of the quotient group $\mathbb{Z}/4\mathbb{Z}$.

A useful way to visualize cosets is:

G = union of disjoint cosets of N

G:
|---- N ----|---- g1N ----|---- g2N ----|---- g3N ----|

Each block has the same number of elements as $N$, and together they cover the whole group.

2. Normal Subgroups and Why They Are Necessary

• A subgroup $N$ must be normal for the factor group operation to be well-defined. Normality means $$gN = Ng \quad \text{for all } g \in G$$ or equivalently $$gNg^{-1} = N \quad \text{for all } g \in G.$$ This ensures that the product of two cosets does not depend on which representatives are chosen.

• If $N$ is not normal, then the expression $(gN)(hN)$ may produce different results depending on the chosen representatives $g$ and $h$. That would make the “multiplication” on cosets inconsistent, so no group structure would exist.

Example:

• In the group $\mathbb{Z}$, every subgroup is normal because the group is abelian.
• In the symmetric group $S_3$, the subgroup $\{e, (12)\}$ is not normal, so one cannot form a factor group $S_3/\{e,(12)\}$.

Normal subgroups are therefore the exact subgroups that allow a group to be divided into a new group.

3. Structure and Properties of the Factor Group

• The factor group $G/N$ inherits a group structure from $G$. Its identity element is the coset $N$ itself, and the inverse of $gN$ is $g^{-1}N$: $$(gN)^{-1} = g^{-1}N.$$ The factor group is often simpler than the original group, because many distinct elements of $G$ are identified as the same coset.

• The size of a factor group depends on the index of the subgroup: $$|G/N| = [G : N]$$ when $G$ is finite. This means the number of elements in the quotient group equals the number of cosets of $N$ in $G$.

Example:

• If $G = \mathbb{Z}$ and $N = n\mathbb{Z}$, then the factor group $\mathbb{Z}/n\mathbb{Z}$ has exactly $n$ elements: $$\{0+n\mathbb{Z}, 1+n\mathbb{Z}, \dots, (n-1)+n\mathbb{Z}\}.$$ This is the group of integers modulo $n$.

Important consequences:

• Factor groups help describe how a group can be decomposed.
• They are central in the First Isomorphism Theorem, which states that many groups are naturally isomorphic to a quotient group.
• They are used to study kernels of homomorphisms, since every kernel is a normal subgroup.

Working / Process

1. Choose a group $G$ and a normal subgroup $N$.

First identify the parent group and verify that the subgroup is normal. Without normality, the quotient construction fails.

2. Form the cosets of $N$ in $G$.

List all distinct left cosets $gN$. These cosets partition the group, and each coset becomes a single element in the factor group.

3. Define multiplication on cosets and simplify.

Multiply cosets using $$(gN)(hN) = (gh)N.$$ Then write the resulting cosets as the elements of the factor group and check the identity and inverses.

Example with $\mathbb{Z}/4\mathbb{Z}$:

• Choose $G = \mathbb{Z}$, $N = 4\mathbb{Z}$.
• The cosets are $0+4\mathbb{Z}, 1+4\mathbb{Z}, 2+4\mathbb{Z}, 3+4\mathbb{Z}$.

• Addition is performed modulo 4: $$(1+4\mathbb{Z}) + (3+4\mathbb{Z}) = 4+4\mathbb{Z} = 0+4\mathbb{Z}.$$

• So the factor group behaves like the integers modulo 4.

Advantages / Applications

• Factor groups simplify complicated groups by compressing elements into equivalence classes, making the structure easier to analyze.
• They are essential in understanding homomorphisms, because the quotient by a kernel produces a group naturally related to the image.
• They are widely used in solving congruences, studying symmetries, constructing new algebraic systems, and proving major theorems in abstract algebra.

Typical applications include:

• Modular arithmetic: $\mathbb{Z}/n\mathbb{Z}$
• Geometry and symmetry groups
• Ring theory and field theory
• Classification of groups and algebraic structures
• Cryptography and coding theory

Summary

• A factor group is formed by dividing a group into cosets of a normal subgroup.
• It turns the set of cosets into a new group under coset multiplication.
• Factor groups are useful because they simplify groups and reveal hidden structure.

Factor group, coset, normal subgroup, quotient group, and kernel are the main terms to remember.