Engineering / Mathematics — Abelian Groups
Abelian Groups: Definitions, Properties and Examples
Abelian groups: axioms, standard examples, and why the commutative case is sufficient for computational number theory.
Executive summary
A group is a set with an associative operation, an identity and inverses. Restricting to the commutative case loses nothing for this subject and simplifies the theory considerably.
The examples that matter are few and recur constantly: the integers under addition, the integers modulo n under addition, and the units modulo n under multiplication.
Learning objectives
- State the group axioms and verify them on examples.
- Distinguish additive from multiplicative notation.
- Identify the groups that recur throughout the subject.
01The axioms
Abelian group
A set G with a binary operation satisfying:
Associativity — (a · b) · c = a · (b · c)
Identity — there is e with a · e = a for all a
Inverses — every a has a⁻¹ with a · a⁻¹ = e
Commutativity — a · b = b · a
The identity and each inverse are unique, both by short arguments from the axioms. Uniqueness of the identity follows by comparing two candidates; uniqueness of inverses follows by associativity.
02Notation
| Concept | Additive | Multiplicative |
|---|---|---|
| Operation | a + b | a · b or ab |
| Identity | 0 | 1 |
| Inverse | −a | a⁻¹ |
| Repetition | na | a^n |
| Subgroup generated | ⟨a⟩ = {na} | ⟨a⟩ = {a^n} |
03The examples that matter
Z under addition
Infinite cyclic, generated by 1. The model for every cyclic group.
Z_n under addition
Cyclic of order n. Discrete logarithms here are trivial — division by the generator modulo n.
Z_n* under multiplication
Order φ(n). The central object of primality testing and discrete logarithm cryptography.
F_q* for a finite field
Cyclic of order q − 1. The setting for finite field discrete logarithms.
The contrast between the second and third is the foundation of public-key cryptography. Both are abelian groups of similar size, but discrete logarithms are trivial in the additive group and believed hard in the multiplicative one. The difficulty lives in the representation, not in the abstract structure.
04Frequently asked questions
Is the closure axiom missing?
It is implicit in calling the operation a binary operation on G, which by definition returns an element of G. Many treatments list it separately for emphasis; nothing changes either way.
Why do discrete logarithms differ in difficulty between isomorphic groups?
Because an isomorphism need not be efficiently computable. Z_n under addition and a cyclic subgroup of Z_p* of order n are isomorphic as abstract groups, and constructing the isomorphism is exactly the discrete logarithm problem.
Are non-abelian groups ever relevant here?
Not in this collection. They appear in braid group cryptography and in the analysis of some algorithms, but every group used for the number-theoretic algorithms treated here is abelian.
Sources and method
Structural reference: Victor Shoup, A Computational Introduction to Number Theory and Algebra, Version 1, Cambridge University Press, 2005 — book pages 180-185.
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.