Mathematics.

group theory

Quotient Groups

Abstract Algebra I35 minDifficulty6 out of 10

You should know: normal subgroups

Overview

Given a group G and a normal subgroup N ⊴ G, the quotient group G/N is the set of cosets of N in G, made into a group by the operation (aN)(bN) = (ab)N. It is well-defined precisely because N is normal, and it formalizes the idea of 'collapsing' N to a single identity element while retaining a coherent group structure on what remains. Quotient groups are central to the first isomorphism theorem, which identifies G/ker(φ) with the image of any homomorphism φ.

Intuition

Forming G/N is like agreeing to treat every element of N as 'the same as' the identity — you zoom out and only track which coset an element belongs to, ignoring differences within a coset. This only produces a consistent group operation when N is normal, because then multiplying representative-by-representative gives the same coset no matter which representatives you pick. A concrete instance everyone already knows is clock arithmetic: ℤ/12ℤ collapses all integers differing by a multiple of 12 into one of 12 classes, exactly matching a 12-hour clock.

Formal Definition

Definition

Let N ⊴ G. The quotient group G/N consists of the left cosets of N in G, with operation and identity:

G/N={gN:g∈G}G/N = \{ gN : g \in G \}
Underlying set
(aN)(bN)=(ab)N(aN)(bN) = (ab)N
Well-defined coset multiplication (requires N normal)
eG/N=N=eNe_{G/N} = N = eN
Identity of the quotient group
∣G/N∣=[G:N]=∣G∣/∣N∣(G finite)|G/N| = [G:N] = |G|/|N| \quad (G \text{ finite})
Order of the quotient

Worked Examples

  1. ℤ is abelian so 4ℤ is automatically normal. The cosets partition ℤ by remainder mod 4.

    Z/4Z={0+4Z, 1+4Z, 2+4Z, 3+4Z}\mathbb{Z}/4\mathbb{Z} = \{0+4\mathbb{Z},\ 1+4\mathbb{Z},\ 2+4\mathbb{Z},\ 3+4\mathbb{Z}\}

  2. Addition of cosets adds representatives mod 4, e.g. (2+4ℤ)+(3+4ℤ) = 5+4ℤ = 1+4ℤ.

    (2+4Z)+(3+4Z)=1+4Z(2+4\mathbb{Z}) + (3+4\mathbb{Z}) = 1 + 4\mathbb{Z}

  3. This is exactly the cyclic group of order 4.

    Z/4Z≅Z4\mathbb{Z}/4\mathbb{Z} \cong \mathbb{Z}_4

Answer: ℤ/4ℤ has 4 elements {0,1,2,3} (mod 4) and is isomorphic to the cyclic group ℤ₄.

Practice Problems

Difficulty 4/10

What is the order of the quotient group ℤ/6ℤ, and list its elements?

Difficulty 5/10

Let G = S₃ (order 6) and N = A₃ = {e, (123), (132)} (order 3, normal since index 2). Compute |G/N| and identify the group.

Difficulty 6/10

Why does coset multiplication (aN)(bN) = (ab)N fail to be well-defined if N is not normal?

Quiz

The quotient group G/N is defined for a normal subgroup N because normality guarantees:
For finite G with N ⊴ G, the order of G/N equals:
ℤ/nℤ, the quotient of the integers by the subgroup of multiples of n, is isomorphic to:

Summary

  • For N ⊴ G, the quotient group G/N consists of cosets gN with operation (aN)(bN) = (ab)N, well-defined exactly because N is normal.
  • |G/N| = [G:N] = |G|/|N| for finite G; ℤ/nℤ is the canonical example, isomorphic to ℤₙ.
  • Quotient groups let you 'collapse' a normal subgroup to a point while preserving a legitimate group structure on the rest.

References

  1. WebsiteWikipedia — Quotient group