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If $G/Z(G)$ is cyclic, then $G$ is abelian. Indeed, if $gZ(G)$ generates the quotient, every element is $g^{a}z$ with $z\in Z(G)$, and any two such elements commute.

The word metric measures how many allowed moves are needed to change one group element into another. For a finite symmetric generating set of a group $S$ with $1\notin S$, it is the graph distance between vertices of the Cayley graph, equivalentlyHere word length counts letters in a shortest representative of $g^{-1}h$. The path metric on the entire Cayley graph also permits points inside edges; the word metric is its restriction to the group itself.^[ref]

The word metric is left invariant:This corresponds to the label-preserving group action on the Cayley graph. Symmetry follows by reversing a shortest graph path; concatenating routes gives the triangle inequality.^[ref]

Right invariance can fail in a non-abelian group, because right multiplication replaces $g^{-1}h$ by $a^{-1}{g}^{-1}ha$ through conjugation. In the free group $F(x,y)$ with $S=\{x,x^{-1},y,y^{-1}\}$, we have $d_S(1,y)=1$, whereas $d_S(x,yx)=|x^{-1}yx{|}_S=3$: the latter freely reduced word has three letters. In an abelian group, conjugation fixes each element, so right invariance does hold. Cayley graphs of the integers and word metric balls in the integer lattice give examples where the word metric can be calculated explicitly.

For an order-$k$ scalar quasilinear partial differential equation with real-analytic coefficients, real-analytic Cauchy data on a real-analytic non-characteristic hypersurface determine a unique real-analytic solution in a neighbourhood of each point of that hypersurface.

For a scalar equation of order $k$, Cauchy data consist of the first $k$ normal jets $u,{\partial }_{N}u,\dots ,{\partial }_N^{k-1}u$ on the initial hypersurface.

These prescribed values define the initial conditions of a Cauchy problem.

For $0<s<a$,

For $a>0$ and real $c$, integrate $e^{-a{z}^2}$ around a wide rectangle between the real axis and $\mathrm{Im}z=c$. The vertical contributions vanish as the width tends to infinity, so

If $f$ is analytic and bounded by $K$ on $\mathrm{Re}z>0$, then Cauchy's derivative estimate giveson $\mathrm{Re}z>c$. Integrating along line segments gives a Lipschitz bound with the same constant in that smaller half-plane.

A functor $F:\mathcal{C}\to \mathcal{D}$ is a discrete fibration when every arrow $B\to FA$ has a unique lift with codomain $A$.

A functor $F:\mathcal{C}\to \mathcal{D}$ is final when every comma category $(B\downarrow F)$ is nonempty and connected. Colimits are unchanged after restriction along a final functor.

The category $\mathbf{Met}$ has metric spaces as objects and maps $f$ satisfying $d(fx,fy)\le d(x,y)$ as morphisms. Its binary product uses the maximum metric.

For $F:\mathcal{C}\to \mathcal{D}$ and $B\in \mathcal{D}$, the objects of $(B\downarrow F)$ are arrows $B\to FA$; its morphisms are commuting triangles induced by arrows in $\mathcal{C}$.

A regular category has finite limits, image factorizations into regular epimorphisms followed by monomorphisms, and pullback-stable regular epimorphisms.

An endomorphism $e:A\to A$ is idempotent when $e^2=e$. It splits when $e=ir$ for maps $r:A\to B$, $i:B\to A$ satisfying $ri=1_B$.

A colimit is a universal cocone under a diagram: every other cocone factors through it uniquely.

A limit of a diagram $D:\mathcal{J}\to \mathcal{C}$ is a terminal cone over $D$: every other cone factors through it uniquely.

A terminal object $1$ of a category receives exactly one morphism $X\to 1$ from every object $X$. Any two terminal objects are uniquely isomorphic by composing their unique connecting maps.

A singleton is terminal in the Category of sets, and the trivial group is terminal among groups. The empty set is not terminal, because no function from a nonempty set to it exists. An object that is both initial and terminal is a zero object; this occurs for the trivial group but not in the category of all sets.

A subobject of an object $A$ in a category is an equivalence class of monomorphisms $m:B\to A$. Two such maps are equivalent if an isomorphism between their domains makes the maps into $A$ agree. This removes dependence on the name of the chosen domain.

In the Category of sets, subobjects correspond to subsets; in the category of groups, they correspond to subgroups. The map into $A$ remains essential: two abstractly isomorphic subgroups can define different subobjects if they occupy different positions in $A$. Dually, quotient objects are equivalence classes of epimorphisms out of an object.

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.

The category of sets has sets as objects and functions as morphisms.

A category is small when its objects and morphisms form sets.