Engineering / Mathematics — Abelian Groups
Quotient Groups
Constructing the quotient group from a subgroup, well-definedness of the induced operation, and the standard examples.
Executive summary
The cosets of a subgroup can themselves be given a group structure, provided the operation on cosets is well defined. In an abelian group it always is.
The construction of the integers modulo n is exactly this quotient, which makes modular arithmetic a special case of a general algebraic operation.
Learning objectives
- Construct the quotient group and verify the operation is well defined.
- Identify Z_n as a quotient of Z.
- State the order of a quotient group.
01The construction
Quotient group
For a subgroup H ≤ G with G abelian, the set of cosets G/H forms a group under (aH)(bH) = (ab)H, with identity H and inverse (aH)⁻¹ = a⁻¹H.
Well-definedness
If aH = a'H and bH = b'H then (ab)H = (a'b')H.
Reason. Write a' = ah₁ and b' = bh₂. Then a'b' = ah₁bh₂ = ab(h₁h₂) using commutativity, and h₁h₂ ∈ H.
02The canonical example
Taking G = Z under addition and H = nZ, the multiples of n, gives cosets that are exactly the residue classes modulo n.
Z / nZ = Z_n, with coset a + nZ corresponding to the residue class of aSo modular arithmetic is not a special trick but an instance of a general construction. The well-definedness of addition modulo n, which had to be checked directly when residue classes were introduced, is the general theorem applied to this case.
The order of the quotient follows from Lagrange: |G/H| = |G|/|H|, matching the count of n residue classes.
03Quotients in the multiplicative setting
| Group G | Subgroup H | Quotient G/H |
|---|---|---|
| Z | nZ | Z_n, order n |
| Z_p* | Quadratic residues | Order 2; the Legendre symbol |
| Z_n* | ⟨a⟩ | Order φ(n)/ord(a) |
| F_q* | Subgroup of order d | Order (q−1)/d |
The second row identifies the Legendre symbol as a quotient map. The quadratic residues form a subgroup of index 2 in Z_p*, and the symbol is precisely the map to the two-element quotient group, which explains why it is multiplicative.
04Frequently asked questions
Why is normality needed in the non-abelian case?
Because without it the product of two cosets need not be a coset, so the operation is not well defined. Normality is exactly the condition making left and right cosets coincide, which is what the well-definedness argument requires.
Is the quotient a subgroup of G?
No, and this is a common confusion. Its elements are cosets, which are subsets of G rather than elements of it. The quotient is a new group built from G, not contained in it.
What does the quotient by the trivial subgroup give?
A group isomorphic to G itself, since each coset is a single element. Quotienting by the whole group gives the trivial group. These extremes bracket the construction.
Sources and method
Structural reference: Victor Shoup, A Computational Introduction to Number Theory and Algebra, Version 1, Cambridge University Press, 2005 — book pages 190-194.
This page carries the durable method layer only: definitions, constructions, algorithms, complexity results and selection criteria, authored originally for KEVOS. No text is transcribed or paraphrased from the source, and no numeric tables or benchmark data are reproduced — these are routed to live authoritative sources instead.
Author: Kevin Jogin. Last reviewed 2026-08-07.