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The class of all finitely generated abelian groups has the closure properties needed for the Hurewicz theorem modulo a Serre class. In a simply connected space, degreewise membership of homotopy groups and integral homology groups in this class is equivalent.

A sheaf of sets is a presheaf in which compatible sections on any open cover glue to one unique section on the union.

A sheaf may additionally carry abelian group or module structure on its sections, with restriction and gluing preserving that structure. The underlying sheaf of sets retains the same local-to-global axiom.

A sheaf of abelian groups is a sheaf of sets whose sections on each open set form an abelian group and whose restriction maps are group homomorphisms. Its stalk of a sheaf is the corresponding group of germs. Kernels are computed on sections, whereas cokernels are obtained by sheafification of the sectionwise cokernel. An exact sequence can therefore be checked on stalks.

The odd integers form an abelian group under the displayed operation. The identity element is $-1$ and the inverse element of $a$ is $-a-2$. The bijection $a\mapsto (a+1)/2$ is a group isomorphism to $(\mathbb{Z},+)$, since it converts the shifted operation into addition. This illustrates how an unfamiliar identity and inverse can arise from ordinary addition by a change of coordinates.

A Sidon set in an abelian group is a subset $A$ for which an equality $a+b=c+d$ with $a,b,c,d\in A$ forces the unordered pairs $\{a,b\}$ and $\{c,d\}$ to be equal. Equivalently, the ordered nonzero differences $a-b$ with $a\ne b$ are all distinct.

A simplicial abelian group is a simplicial object in abelian groups. Its underlying simplicial set is always a Kan complex.

The singular chain group is the free abelian group on all continuous maps ${\Delta }^q\to X$, the singular simplices. In particular it is free even when the set of simplices is infinite. This freeness makes applying $\mathrm{Hom}(C_q(X),-)$ to a short exact coefficient sequence exact.

A singular $n$-cochain with coefficients in an abelian group $A$ is a homomorphism from the singular chain group $C_n(X;\mathbb{Z})$ to $A$. It is specified by its values on all singular simplices, without a finite-support restriction. Its coboundary evaluates on the alternating sum of the faces. The resulting cochain complex computes singular cohomology.

For a topological space $X$ and abelian group $A$, singular cohomology is the cohomology of the cochain complex $C^n(X;A)=\mathrm{Hom}(C_n(X;\mathbb{Z}),A)$, where $C_n$ is the singular chain group. Its differential is dual to the singular boundary. This theory has relative cohomology, long exact sequences and the Excision theorem. On spaces that are not locally contractible it can differ from Čech cohomology.

For $A$ in a finite abelian group and normalized Fourier transform $\widehat{1_A}$, the normalized number of sextuples satisfyingisThis follows by inserting orthogonality of complex exponentials for the displayed equation: the three variables on one side contribute $\widehat{1_A}(\gamma {)}^3$ and those on the other side contribute its complex conjugate.

For a point inclusion $i:\{x\}\hookrightarrow X$ and an abelian group $A$, the skyscraper sheaf $i_{\ast }A$ has sections $A$ on opens containing $x$ and zero on other opens. Its restriction maps are identities or maps to zero. If $A\ne 0$, its nonzero stalks occur at every point of the closure $\overline{\{x\}}$: exactly those points whose every neighbourhood contains $x$. Thus the familiar assertion that only the stalk at $x$ is nonzero requires $x$ to be a closed point. It is always flasque, giving the stated skyscraper sheaf cohomology.

The assertion follows using only Sylow theorems. Write the order instead as $p{q}^2$ to separate the prime of multiplicity one. If $p<q$, the number of Sylow $q$-subgroups divides $p$ and is one modulo $q$, so one such subgroup is normal. If $p>q$ and the Sylow $p$-subgroup is not normal, its number must be $q^2$, forcing $p\mid (q-1)(q+1)$. Since $p>q$, this forces $p=q+1$ and hence $p=3,q=2$. In that order-twelve exception, four order-three subgroups account for eight nonidentity elements. The four remaining elements contain every order-four Sylow subgroup, so that subgroup is unique and normal. In every case a normal Sylow subgroup and its quotient have prime or prime-square order, hence are abelian groups. They provide a series with abelian factors and prove solubility. Orders with the squared exponent lowered to one or zero follow by the same simpler counting.

The square-class group identifies two nonzero elements of a field when their quotient is a square. Multiplication makes it an abelian group of exponent two, hence a vector space over ${\mathbb{F}}_2$. It records quadratic extensions and the images of Kummer theory for $n=2$.

Every subgroup of an abelian group is normal. Hence every intermediate field of a finite abelian Galois extension is itself Galois over the base field.

Every subgroup of an Abelian group generated by $n$ elements has at most $n$ generators. Lift it under a surjection ${\mathbb{Z}}^n\to A$. A subgroup $L\le {\mathbb{Z}}^n$ has at most $n$ generators by induction: its first-coordinate image is either zero or $d\mathbb{Z}$. In the latter case choose a lift of $d$, and subtract its multiples to reduce to the kernel in ${\mathbb{Z}}^{n-1}$. The images of these generators generate the original subgroup. The Abelian hypothesis matters: $S_{2n+2}$ is generated by a transposition and a full cycle but contains the elementary abelian group generated by $n+1$ disjoint transpositions, requiring $n+1$ generators by its vector-space dimension over ${\mathbb{F}}_2$.

A sum combines a finite or infinite sequence of terms by repeated addition. The symbol $\sum$ denotes summation over an index.

For finite sets $A,B\subseteq G$ in an abelian group $G$, their sumset isEach distinct value occurs only once in this set, even if many ordered pairs produce it. When $B=A$, repetitions are allowed: $a+a$ belongs to $A+A$. For the two integer sets introduced above,For $A$, the value $3$ comes from $(0,3),(1,2),(2,1),(3,0)$. For $B$, two ordered pairs give the same sum only when one reverses the other, apart from a pair being identical to itself. Thus its four pairs $(b,b)$ and six pairs with distinct members, counted without regard to order, produce ten distinct sums.

Every nonempty finite set of integers $S\subseteq \mathbb{Z}$ satisfies $|S+S|\ge 2|S|-1$. To see this, list its members as $s_1<\cdots <s_n$. The $2n-1$ distinct sumsall belong to $S+S$. A nonconstant arithmetic progression attains this bound. The ordered list is essential: in the cyclic group $\mathbb{Z}/6\mathbb{Z}$, the subgroup $H=\{0,2,4\}$ instead satisfies $H+H=H$, of cardinality three. Addition wraps around, for example $4+4=2$ in this group.

For finite sets $A,B\subseteq G$ in an abelian group $G$, the sumset representation function isIt counts ordered pairs in the Cartesian product, so $(a,b)$ and $(b,a)$ are different when $a\ne b$, even though commutativity makes their sums equal. It is zero outside the sumset, andFor $A=\{0,1,2,3\}$, its values at $s=0,1,\dots ,6$ are $1,2,3,4,3,2,1$. For $B=\{0,1,4,10\}$, its values at the increasing elements of $B+B$ are $1,2,1,2,2,1,2,2,2,1$. Both lists total sixteen, but the first concentrates more ordered pairs on fewer sums. This distinction is what additive energy quantifies.

The tensor product of an $R$-module $M$ and an $R$-module $N$ is an $R$-module $M{\otimes }_{R}N$ generated by pure tensors $m\otimes n$, subject to additivity in both variables and the balancing relation

A topological abelian group $A$ with a continuous action of a topological group $G$ by group automorphisms. Continuity means joint continuity of $G\times A\to A$. For a discrete coefficient group this is equivalent to every element of $A$ having an open stabilizer.