Definition of a Quotient Group

A quotient group, denoted as $ G/N $, is a group constructed from a group $ G $ and a normal subgroup $ N $ of $ G $. The quotient group consists of the equivalence classes of elements in $ G $ under the equivalence relation defined by $ a \equiv b $ if and only if $ a^{-1}b \in N $. This construction allows the study of the structure of $ G $ by "factoring out" the elements of $ N $, effectively collapsing the group into a smaller group whose structure reflects the original group's properties modulo the normal subgroup.

The definition of a quotient group relies on the existence of a normal subgroup $ N $, which must satisfy two critical conditions:

1. Normality: For all $ g \in G $, $ gNg^{-1} = N $.
2. Compatibility: The equivalence relation $ a \equiv b $ if $ a^{-1}b \in N $ is compatible with the group operation.

The quotient group $ G/N $ is then defined as the set of cosets $ {aN \mid a \in G} $, with the operation $ (aN)(bN) = (ab)N $. This operation is well-defined because $ a^{-1}b \in N $ implies $ (aN)(bN) = a(bN) = (ab)N $, ensuring consistency across all cosets.

Construction Logic of a Quotient Group

The construction of a quotient group is a fundamental operation in group theory, rooted in the concept of equivalence classes. Given a group $ G $ and a normal subgroup $ N $, the quotient group $ G/N $ is formed by partitioning $ G $ into disjoint subsets called cosets, where each coset is of the form $ aN $ for $ a \in G $. These cosets are grouped into equivalence classes based on the relation $ a \equiv b $ if $ a^{-1}b \in N $.

The process of constructing $ G/N $ involves three key steps:

1. Partitioning: The elements of $ G $ are partitioned into cosets of $ N $.
2. Defining the Operation: The group operation on $ G/N $ is defined by $ (aN)(bN) = (ab)N $.
3. Verifying Closure: The resulting set $ G/N $ is closed under the defined operation, and the operation is compatible with the equivalence relation.

The normality of $ N $ ensures that the equivalence classes are invariant under conjugation, making $ G/N $ a well-defined group. For example, consider the group $ \mathbb{Z}/n\mathbb{Z} $, where $ n \in \mathbb{N} $, and $ N = n\mathbb{Z} $. The quotient group $ \mathbb{Z}/n\mathbb{Z} $ consists of the equivalence classes $ {0 + k n \mid k \in \mathbb{Z}} $, forming a cyclic group of order $ n $.

Order of a Quotient Group

The order of a quotient group $ G/N $ is a crucial property that quantifies the size of the resulting group. If $ |G| $ denotes the order of $ G $ and $ |N| $ denotes the order of $ N $, then the order of $ G/N $ is given by the index $ [G:N] = |G| / |N| $. This result is a direct consequence of the first isomorphism theorem, which states that $ G/N \cong G/N $, and the order of the quotient group is equal to the index of the normal subgroup.

The theorem guarantees that the quotient group's order is an integer, and it is only defined when $ N $ is a normal subgroup of $ G $. For instance, in the symmetric group $ S_3 $, the normal subgroup $ A_3 $ has order 3,