Abelian Groups — Mathematics Knowledge System

Overview

An abelian (or commutative) group is a group in which the operation commutes: a·b = b·a for all elements a, b. Named after Niels Henrik Abel for his work on the solvability of polynomial equations, abelian groups are structurally much simpler than general groups and are completely classified: the Fundamental Theorem of Finite Abelian Groups says every finite abelian group is a direct product of cyclic groups of prime-power order, unique up to reordering. Familiar examples include the integers under addition, ℤₙ, and vector spaces under addition.

Intuition

In an abelian group, order of operations never matters — 'do A then B' and 'do B then A' land you in the same place, unlike, say, rotating and then flipping a book versus flipping then rotating. This extra symmetry is what makes abelian groups so much easier to fully classify: instead of a zoo of possible structures, every finite one is built by gluing together independent cyclic 'clocks' of prime-power length, exactly the direct-product decomposition used for ℤₙ. The commutativity is also exactly why every subgroup of an abelian group is automatically normal — conjugation gag⁻¹ = gg⁻¹a = a does nothing.

Formal Definition

Definition

A group (G, ·) is abelian if it additionally satisfies commutativity:

a \cdot b = b \cdot a \quad \text{for all } a, b \in G

Commutativity axiom

G \cong \mathbb{Z}_{p_1^{e_1}} \times \mathbb{Z}_{p_2^{e_2}} \times \cdots \times \mathbb{Z}_{p_k^{e_k}}

Fundamental Theorem of Finite Abelian Groups (invariant factor / prime-power form)

(ab)^n = a^n b^n \quad \text{for all } n \in \mathbb{Z}

Power of a product (only valid because operation commutes)

\text{Every subgroup of an abelian group is normal.}

Automatic normality

Worked Examples

Addition of integers mod 6 commutes: a+b ≡ b+a (mod 6) trivially, since ordinary integer addition commutes.
$a+b \equiv b+a \pmod 6$
By Lagrange, subgroup orders divide 6: possible orders 1, 2, 3, 6.
$\text{orders} \in \{1,2,3,6\}$
The actual subgroups are {0} (order 1), {0,3} (order 2), {0,2,4} (order 3), and ℤ₆ itself (order 6) — one for each divisor.
$\{0\},\ \{0,3\},\ \{0,2,4\},\ \mathbb{Z}_6$

Answer: ℤ₆ is abelian with exactly 4 subgroups, one for each divisor of 6: orders 1, 2, 3, 6.

Practice Problems

Difficulty 4/10

How many abelian groups of order 12 are there up to isomorphism?

Difficulty 5/10

Show that in an abelian group, (ab)ⁿ = aⁿbⁿ for all n ≥ 1, and explain why this can fail in a non-abelian group.

Difficulty 5/10

Explain why every subgroup of an abelian group is normal, and give the practical consequence for forming quotient groups.

Quiz

Summary

An abelian group satisfies a·b = b·a for all elements; every subgroup is automatically normal.
The Fundamental Theorem of Finite Abelian Groups classifies every finite abelian group as a direct product of cyclic groups of prime-power order.
There are exactly p(k) isomorphism classes of abelian groups of order pᵏ, one for each integer partition of k (e.g. 3 classes for order 8 = 2³).

References

WebsiteWikipedia — Abelian group