Finitely generated abelian group

A finitely generated abelian group has a finite generating set. Equivalently, it is the cokernel of a homomorphism between finitely generated free abelian groups.

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Finite generation detected by a prime quotient ( $A finite ⟺ A / p A = 0 for some prime p$ )

The Fundamental theorem of finitely generated abelian groups writes

A = Z^{r} \oplus F

with

F

finite. Reduction modulo a prime contains

(Z / p Z)^{r}

, so a vanishing quotient forces

r = 0

. Conversely multiplication by a prime coprime to

∣ F ∣

is bijective on

F

. Finite generation matters: the nonzero divisible group

(Q, +)

satisfies

p Q = Q

for every prime and is not finitely generated, since any finite list of rational generators has a common denominator.

Rank of an abelian group

The rank of a finitely generated abelian group

G

is the number

r

of infinite cyclic factors in its invariant-factor decomposition, equivalently

dim_{Q} (G \otimes_{Z} Q)

.

Fundamental theorem of finitely generated abelian groups

Every finitely generated abelian group is isomorphic to

Z^{r} \oplus Z / d_{1} Z \oplus \dots \oplus Z / d_{s} Z, d_{1} ∣ d_{2} ∣ \dots ∣ d_{s} .

The rank

r

and invariant factors

d_{i} > 1

are unique.

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Finitely generated abelian group by

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A finitely generated abelian group is a specific type of group in abstract algebra that has some important properties. 1. **Group**: An abelian group is a set equipped with an operation (often called addition) that satisfies four properties: closure, associativity, identity, and inverses. Additionally, an abelian group is commutative, meaning that the order in which you combine elements does not matter (i.e.

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