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An abelian extension is a Galois extension whose Galois group is an abelian group.

An abelian progression is a generalized arithmetic progression inside an abelian group. It is the image of an integer box under a group homomorphism.

An abelian quotient is a quotient group $G/N$ that is an abelian group. It exists only for a normal subgroup $N$, and it is abelian exactly when $N$ contains the commutator subgroup. Thus every abelian quotient factors through the abelianization $G/[G,G]$.

An abelian subgroup is a subgroup that is an abelian group under the restricted operation.

Additive combinatorics studies how the sizes and representation functions of sumsets reveal algebraic structure in subsets of abelian groups.

For a set of shifts $H$ and a map $\theta$ into a circle group, this additive energy counts quadruples whose shift sums and frequency sums both agree. It is the additive energy of the graph of $\theta$ in the product abelian group.

An additive group is an abelian group whose operation is written as addition, with identity $0$ and inverse of $x$ written $-x$.

If $G$ is a finite abelian group of odd order $N$ and $A\subseteq G$ contains no nonconstant three-term arithmetic progression, then Bourgain's bound is

For a linear map $T:V\to W$, the cokernel is the quotient $\mathrm{coker}T=W/\mathrm{im}T$. The same definition applies to a homomorphism of abelian groups.

A commutative monoid is a monoid whose operation is commutative. The natural numbers with addition form a commutative monoid that is not an abelian group, because positive elements have no additive inverses.

For a nonnegative integer $d$ and an abelian group $A$,The one-point compactification is the sphere $S^d$, and it is locally contractible at the added point. Apply the compact-support comparison with a one-point compactification and the reduced cohomology of a sphere. In dimension zero the compactification is $S^0$, giving the same formula.

A coset progression is a sumset $H+P$, where $H$ is a finite subgroup of an abelian group and $P$ is a generalized arithmetic progression.

A direct system indexed by a directed set $A$ consists of abelian groups $G_a$ and homomorphisms ${\varphi }_{ab}:G_a\to G_b$ for $a\le b$, with ${\varphi }_{aa}=1$ and ${\varphi }_{bc}{\varphi }_{ab}={\varphi }_{ac}$.

A divisible group is an abelian group $G$ in which, for every positive integer $n$ and every $g\in G$, some $h\in G$ satisfies $nh=g$.

A connected Eilenberg–MacLane space $K(G,n)$, $n\ge 1$, has ${\pi }_n\cong G$ and all other positive-degree homotopy groups trivial. The group $G$ must be abelian when $n\ge 2$, but may be nonabelian when $n=1$. For an abelian group, the Dold–Kan correspondence constructs a simplicial model from the chain complex concentrated in degree $n$. A classifying space of a discrete group gives the general $K(G,1)$ case.

An elementary abelian $p$-group is a finite abelian group in which every nonidentity element has order $p$. It is equivalently a finite-dimensional vector space over ${\mathbb{F}}_p$ under addition.

For finitely supported random variables $A,B$ in an abelian group,To prove it, take two conditionally independent copies $(A_1,B_1)$ and $(A_2,B_2)$ of $(A,B)$ given $A+B$. Since $A_1+B_1=A_2+B_2$, entropy submodularity givesThe first two terms are at most $H(A)+H(B)$. The last joint entropy isConsequently $H(A_1-B_2)\le H(A+B)+2I(A;B)$. Subtractingfrom this bound proves the theorem.

An epimorphism $e:A\to B$ is right-cancellative: $fe=ge$ implies $f=g$. In the Category of sets and in abelian groups, epimorphisms are precisely the surjective homomorphisms.

A finite CW complex has finitely many cells. Its cellular chain complex consists of finitely generated free abelian groups and vanishes outside finitely many degrees, so its integral homology groups are finitely generated.

Every finitely generated abelian group is isomorphic toThe rank $r$ and invariant factors $d_i>1$ are unique.