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An abelian group of order $p^2$ is cyclic if it has an element of order $p^2$. Otherwise Lagrange's theorem makes every nonidentity element have order $p$. Choose $a\ne 1$ and $b\notin ⟨a⟩$. The group homomorphism $(i,j)\mapsto a^i{b}^j$ from $C_p\times C_p$ is injective: $b^j\in ⟨a⟩$ with $j\ne 0$ would imply $b\in ⟨a⟩$, using the inverse of $j$ modulo $p$. Equal finite orders then make this an isomorphism.

A classical variety is affine if finitely many global regular functions generate the unit ideal and their nonvanishing opens are affine. Localization of global sections on a principal open identifies their coordinate rings with $A_{f_i}$, where $A=\Gamma (X,{\mathcal{O}}_X)$. Choose a finite-type subalgebra of $A$ containing the functions, their unit-ideal coefficients and numerators of generators of every $A_{f_i}$. Its affine variety has the same principal affine charts, whose isomorphisms glue globally. This avoids assuming finite generation of $A$ before affineness is known.

For an algebraic group $H$, a flat $H$-torsor over $Y$ is a faithfully flat morphism $X\to Y$ with a right $H$-action such that $X\times H\to X{\times }_{Y}X$, $(x,h)\mapsto (x,xh)$, is an isomorphism. A Zariski torsor additionally becomes $U\times H$ on a Zariski open cover of $Y$.

An algebraic structure is a set equipped with specified algebraic operations satisfying selected axioms. An operation's domain is part of the structure: a binary operation on $S$ takes two elements of $S$ and produces another element of $S$. Groups specify one such operation, rings specify addition and multiplication, and vector spaces specify vector addition and scalar multiplication by a separate field.

Different choices of operations can give the same underlying set different structures. The integers form a group under addition, since every integer has an additive inverse. They do not form a group under multiplication: for example, $2$ has no multiplicative inverse among the integers. Even the nonzero integers fail that inverse requirement. In contrast, nonzero rational numbers form a group under multiplication.

Finite multiplication tables make such distinctions visible. Arithmetic modulo $5$ gives a field; arithmetic modulo $6$ gives a commutative ring with zero divisors. In the former, multiplying by any nonzero element permutes all residue classes. In the latter, multiplying by $2$ reaches only $0,2,4$, and $2\cdot 3=0$ even though both factors are nonzero.

The goal is to extract reusable properties rather than memorize isolated tables. A homomorphism preserves the specified operations, and an isomorphism shows that two structures have identical operation patterns after relabeling.

Algebraic topology translates questions about topological spaces and continuous maps into questions about groups and other structures from algebra. A topological invariant compresses the enormous amount of data in a space into something one can calculate. For example, the fundamental group records which based loops can be continuously deformed into one another. A filled disk has trivial fundamental group, whereas a circle has fundamental group $\mathbb{Z}$: an integer remembers how many times a loop goes around.

This compression also respects maps. By functoriality of the fundamental group, a continuous map preserving the chosen basepoints induces a group homomorphism; successive maps induce successive group homomorphisms. Thus geometry supplies arithmetic constraints. A proposed homotopy equivalence must induce an isomorphism, so incompatible fundamental groups rule it out before anyone attempts to construct the deformation.

The beauty is that a whole annulus can retain exactly the same homotopy type as its inner circle. Its thickness disappears under a deformation retraction, while the obstruction to shrinking a loop survives. Equal fundamental groups alone do not establish a homeomorphism or a homotopy equivalence between arbitrary topological spaces.

This turns a visual question into a reusable test. Suppose a continuous map sends the inner circle of an annulus once around another circle. Could it extend over a filled disk spanning that boundary? Such an extension would supply a homotopy shrinking the image loop to a point. Its winding number would have to change from one to zero, which homotopy invariance of winding number forbids. Algebra rules out the extension even when no particular formula for it has been proposed.

An algebra isomorphism is a bijective linear map preserving multiplication and identity between two algebras over the same field. In classification of a square-zero extension of an algebra, an equivalence must additionally induce the identity on the prescribed quotient and kernel.

For a complex torus $X=V/\Lambda$, every holomorphic line bundle is classified by a pair $(H,\alpha )$. Here $H$ is a Hermitian form, linear in its first argument, whose imaginary part $E$ is integral on $\Lambda \times \Lambda$, and $\alpha$ is a semicharacter of a complex lattice:The corresponding bundle $L(H,\alpha )$ is the quotient of $V\times \mathbb{C}$ by the group actionThe semicharacter identity makes successive lattice translations compose correctly. The resulting correspondence is an isomorphism onto the Picard group, so it gives both existence and uniqueness of the classification.^[ref]

Tensor products add the Hermitian forms and multiply the semicharacters:When $H=0$, the semicharacter is an ordinary unitary group homomorphism, describing bundles with zero First Chern class. A positive-definite $H$ instead gives a polarization of a complex torus. The theorem therefore separates the discrete integral curvature data from the remaining flat holomorphic choices.

For a regular local ring of embedding dimension $e$, a minimal maximal-ideal generating set induces the displayed polynomial-algebra isomorphism. It is surjective in every degree. A nonzero homogeneous kernel relation would bound the cumulative Hilbert function by $O(t^{e-1})$, contradicting the prime-chain lower bound for local length at $\dim R=e$. The graded ring is a domain; the Krull intersection theorem then shows the original ring is a domain by multiplying nonzero initial forms.

The associated subgroups are the source and target $A,B\le G$ of the isomorphism used to define an HNN extension. They determine its pinches in an HNN extension, its coset transversals, and the possible cancellations of adjacent inverse stable letters.

For a finite-dimensional algebra, the Auslander–Reiten quiver has one vertex for each isomorphism class of finite-dimensional indecomposable modules, and its arrows record irreducible module homomorphisms. A homomorphism is irreducible when it is neither a split injection nor a split surjection and every factorization forces the first map to split injectively or the second to split surjectively.

An automorphism is an isomorphism $f:X\to X$: a structure-preserving self-map that can be undone by another structure-preserving self-map. Equivalently, it is an invertible endomorphism. Its inverse satisfieswhere $1_X$ is the identity morphism. Both maps must be morphisms in the chosen category. For ordinary groups, rings, and vector spaces over a fixed field, a bijective morphism automatically has a compatible inverse. For topological spaces, an automorphism is a homeomorphism from the space to itself: a continuous bijection needs a continuous inverse as well. See Riehl, Category Theory in Context, Definition 1.1.10 and Example 1.1.11.

For example, $T(x,y)=(x+y,y)$ is an automorphism of the vector space ${\mathbb{R}}^2$ over the real numbers, with inverse $T^{-1}(u,v)=(u-v,v)$. Both are linear maps. The linear endomorphism $P(x,y)=(x,0)$ from endomorphism preserves addition and scalar multiplication too, but discards a coordinate and cannot be undone. Preserving the structure and preserving enough information for reversal are separate requirements.

The automorphisms of a fixed object in a locally small category form a group under composition in a category: the identity morphism is the identity element, and the inverse element of a composite reverses the order of its factors. This makes reversible internal changes calculable. For a group, the resulting automorphism group records which relabelings preserve all products; the examples in group automorphism show how restrictive that condition can be.

The automorphism group of a group $G$ is the group of all isomorphisms $G\to G$ under composition.

With long-first simple roots ${\alpha }_1={\epsilon }_1-{\epsilon }_2$, ${\alpha }_2={\epsilon }_2$, the Adjoint representation has highest weight ${\epsilon }_1+{\epsilon }_2=2{\Lambda }_2$. Its crystal basis has one vertex $x_{\beta }$ for every root and two distinct zero-weight vertices $z_1,z_2$. The color-one strings are $x_{{\epsilon }_1}\to x_{{\epsilon }_2}$, $x_{{\alpha }_1}\to z_1\to x_{-{\alpha }_1}$, and $x_{-{\epsilon }_2}\to x_{-{\epsilon }_1}$. The color-two strings are $x_{{\epsilon }_1+{\epsilon }_2}\to x_{{\epsilon }_1}\to x_{{\epsilon }_1-{\epsilon }_2}$, $x_{{\epsilon }_2}\to z_2\to x_{-{\epsilon }_2}$, and $x_{-{\epsilon }_1+{\epsilon }_2}\to x_{-{\epsilon }_1}\to x_{-{\epsilon }_1-{\epsilon }_2}$. All other edges vanish. One construction takes the component generated by $1\otimes 2$ in the tensor square of the crystal of the defining odd-orthogonal representation; the crystal tensor-product rule produces exactly these ten vertices. The equivariant isomorphism ${\Lambda }^2{\mathbb{C}}^5\cong {\mathfrak{so}}_5$ identifies its highest weight and representation.

For isomorphic finite-dimensional normed vector spaces, the Banach-Mazur distance is the infimum of $\Vert S\Vert \Vert S^{-1}\Vert$ over their linear isomorphisms. If an $N$-dimensional subspace $E$ of a Banach space of cotype 2 satisfies the summing estimate ${\pi }_2(T)\le L\Vert T\Vert$ for operators from ${\ell }_{\infty }^N$, then $d(E,{\ell }_{\infty }^N)\ge \sqrt{N}/L$: compose an isomorphism with the inclusion and use ${\pi }_2(I_E)=\sqrt{N}$.

Given a pullback in a category $A=B{\times }_{D}C$ and a morphism $D^{\prime }\to D$, assume the indicated pullbacks in a category exist. The four base-changed objects form a pullback in a category. A compatible pair of morphisms into $B{\times }_D{D}^{\prime }$ and $C{\times }_D{D}^{\prime }$ gives morphisms into $B,C$ and a common morphism into $D^{\prime }$. The first two morphisms factor uniquely through $A$, and their common image in $D$ permits unique factorization through $A{\times }_D{D}^{\prime }$. This proves the displayed canonical isomorphism without an assumption that the whole category is complete.

For an algebraic dual space $X^{\ast }={\mathrm{Hom}}_F(X,F)$, the linear map $J:X\to X^{\ast \ast }$ sends $x$ to evaluation at $x$. It is injective because linear functionals separate nonzero vectors: extend any nonzero vector to a basis and use its coordinate functional. If $X$ has a finite basis $e_i$ with dual basis $e_i^{\ast }$, every $B\in X^{\ast \ast }$ equals $J({\sum }_{i}B(e_i^{\ast })e_i)$, so $J$ is an isomorphism. For infinite-dimensional spaces, surjection does not follow from this finite sum argument.

The morphism ${\mathbb{A}}^1\to V(x^4-y^3)$ given by $t\mapsto (t^3,t^4)$ is bijective on points over an algebraically closed field, but is not an isomorphism. At a nonzero curve point, $t=y/x$ is the unique inverse parameter, while the origin has unique preimage zero. The induced map on coordinate rings has image $k[t^3,t^4]\subset k[t]$, which lacks the monomial $t$: all its monomial exponents have the form $3a+4b$, with $a,b\ge 0$. Nor can $t$ be regular at the cusp, since an equality $t=P(t^3,t^4)/Q(t^3,t^4)$ with $Q(0)\ne 0$ would produce a nonzero degree-one term in $tQ$ and none in $P$. Thus pointwise bijectivity does not guarantee an algebraic inverse, even though the inverse is regular away from one singular point.

A rational map with a rational inverse restricts to an isomorphism of dense open subsets. It may identify or omit exceptional points outside those subsets.

The projective morphism ${\mathrm{Proj}}_X{⨁}_{j\ge 0}{\mathcal{I}}^j\to X$ associated to the powers of the ideal sheaf of a closed centre. Off that centre it is an isomorphism. Blowing up a smooth point of an $n$-dimensional smooth variety replaces the point by a projective space ${\mathbb{P}}^{n-1}$, the algebraic exceptional divisor.

A nonzero module is a brick when every endomorphism is either zero or an isomorphism. Equivalently, its endomorphism ring is a division ring.