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If every element of a group squares to the identity, every element equals its inverse. Then $ab=(ab{)}^{-1}=b^{-1}{a}^{-1}=ba$, so the group is an abelian group. The argument does not require a finite group.

A homology group is one of the abelian groups produced by a homology theory. Its elements are homology classes, represented in singular homology by cycles modulo boundaries.

The $n$th homotopy group consists of based homotopy classes of maps $S^n\to X$. It is a group for $n\ge 1$ and an abelian group for $n\ge 2$.

Moving the basepoint around a loop acts on a higher homotopy group. Since ${\pi }_n(X)$ is an abelian group for $n\ge 2$, this action extends linearly to its integral group ring, making ${\pi }_n(X)$ a module. For a connected CW complex, its universal cover gives an equivalent description by deck transformations, together with change of basepoint upstairs. Left and right conventions are converted by inversion of the loop. The Hurewicz theorem respects this action whenever it identifies a higher homotopy group with the corresponding homology of the cover.

Let $L$ be the complexification of a real vector bundle of the tautological bundle over Real projective space ${\mathbb{RP}}^{2n+1}$. For $k\ge 1$, the K-theory Euler class of $L^{\oplus k}$ is $-2^{k-1}([L]-1)$, andThe K-theory Gysin sequence of a sphere bundle gives an extension by $\mathbb{Z}$, which splits as an exact sequence of abelian groups.

A linear phase on an abelian group is an additive character, such as $x\mapsto e(\theta x)$ on the integers or $x\mapsto {\omega }^{rx}$ on ${\mathbb{Z}}_N$.

A long exact sequence is a sequence of abelian groups or modules with arbitrarily many consecutive maps, for which the image of every map equals the kernel of the next. A short exact sequence of cochain complexes yields a long exact sequence of their cohomology, with connecting homomorphisms increasing degree by one. The exact sequences of relative cohomology and the Bockstein homomorphism are examples.

For $f:G\to M_n(\mathbb{C})$ on a finite group, use uniform expectation over the solutions of $x{y}^{-1}z{w}^{-1}=e$ and the ordinary, unnormalized trace to define the displayed fourth-order quantity. If $T_{\rho }$ is its matrix Fourier block, this quantity equals ${\sum }_{\rho }{d}_{\rho }\mathrm{Tr}((T_{\rho }{T}_{\rho }^{\ast }{)}^2)$, so it is real and nonnegative. For a scalar function on an abelian group, it reduces to the usual fourth power of the Gowers uniformity norm of order two. The unnormalized trace means a unitary matrix-valued function can have fourth-order quantity as large as $n$.

A group is metabelian when its commutator subgroup is abelian, equivalently when its derived length is at most two.

For an abelian group $G$ and an integer $m\ge 1$, a Moore space $M(G,m)$ is a CW complex whose reduced homology is $G$ in degree $m$ and zero in every other degree.

The nilpotency class of a nilpotent group $G$ is the least nonnegative integer $s$ for which ${\gamma }_{s+1}(G)=\{1\}$ in the lower central series. Thus nontrivial abelian groups have class one.

A free product $A\ast B$ is nontrivial as a splitting when both factors differ from the trivial group. Alternating words $(ab{)}^r$ with $a,b\ne 1$ in different factors show that every such splitting gives an infinite group. The distinct normal forms $ab$ and $ba$ also show that the group is not an abelian group.

An order in a number field $K$ is a subring $A\subseteq {\mathcal{O}}_K$ containing $1$ and having full rank $[K:\mathbb{Q}]$ as an additive abelian group. Its finite index in the ring of integers of a number field is related to its discriminant by the discriminant-index formula for an integral lattice.

For reduced residue classes $a,n$ modulo $q$,This is the orthogonality relation for the characters of the finite abelian group $(\mathbb{Z}/q\mathbb{Z}{)}^{\times }$.

For the Fourier transform on a finite group, the displayed identity uses uniform expectation in the original function space and a weighted Hilbert-Schmidt inner product in the matrix components. In particular $\Vert f{\Vert }_2^2={\sum }_{\rho }{d}_{\rho }\Vert \hat{f}(\rho ){\Vert }_{\mathrm{HS}}^2$. For an abelian group, all $d_{\rho }=1$, and the identity becomes a sum of squared scalar coefficients.