Glossary of Abstract Algebra
 Level: 0 1 2 3 4 5 6 7
MSC Classification: 8 (12,13,15,16,17,18) (General algebraic systems)

This glossary lists common terms in abstract algebra, at a level typical for graduate school qualifying exams.

 Rough Guides to Algebra
 General Algebra Special Topics

# Linear Algebra

See the article on linear algebra for more details.

Basis
Given some subspace, a basis is a smallest set of elements which generate that subspace; its size is unique, called the dimension of the subspace.
Rank
The rank of a matrix is the dimension of its column space (or row space).
Null Space
The null space of a matrix M is the subspace K with $M\mathbf{v}=0$ for all $\mathbf{v}\in K$. The rank of a matrix plus the dimension of the null space equals the dimension of the overall space.
Similar Matrices
Matrices A and B are similar if there exists an invertible matrix P with $A=PBP^{-1}$; this basically means they're the same up to a change-of-basis.
Trace
If the elements of a square matrix M are represented by $m_{ij}$, the trace of M is $\mathrm{tr}(M)=\sum_{i=1}^n m_{ii}$, the sum of the diagonal elements. The trace is invariant under change-of-basis transformations.
Determinant
The determinant of a square matrix M is a signed summation of matrix entries, with the summation indexed by permutations. The precise formula is $\det(M)=\sum_{\sigma\in S_n} m_{1\sigma(1)} m_{2\sigma(2)}\cdots m_{n\sigma(n)}$. Originally used to "determine" whether a system of linear equations has a solution. Nonzero if and only if the matrix is invertible.
Eigenvalue
The eigenvalue of a square matrix M is a value $\lambda$ such that $M\mathbf{x}=\lambda \mathbf{x}$ for some $\mathbf{x}$, which is called an eigenvector.

# Groups

See the article on beginning abstract algebra for more details.

## Basics

Group
A group is a set G together with an operation $\cdot\:$ that satisfies (i) associativity: $a\cdot(b\cdot c)=(a\cdot b)\cdot c$ for all a,b,c; (ii) existence of an identity: $\exists e\in G$ such that $a\cdot e=e\cdot a=a$ for all a; and (iii) existence of inverses: for any a there exists $a^{-1}$ such that $a\cdot a^{-1}=a^{-1}\cdot a=e$.
Abelian group
A group is abelian if its operation is also commutative: $a\cdot b=b\cdot a$ for any a,b.
Homomorphism
A function $f:(G,\diamondsuit)\to(H,\heartsuit)$ is a homomorphism if $f(g_1\diamondsuit g_2)=f(g_1)\heartsuit f(g_2)$ for all $g,h\in G$.
Isomorphism
A bijective function $f:(G,\diamondsuit)\to(H,\heartsuit)$ is an isomorphism if both f and f-1 are homomorphisms. If the function maps $(G,\diamondsuit)$ to itself, it is an automorphism.
Subgroup, $H<G$
A subset H of a group G that is itself a group is called a subgroup, denoted $H<G$.
Index of a subgroup, $G:H$
The index $G:H$ of a subgroup $H<G$ is the number of cosets of H in G, equal to $|G|/|H|$ by Lagrange's Theorem.
Normal subgroup, $K\lhd G$
A subgroup $K\leq G$ is normal if $gkg^{-1}\in K$ for all $g\in G$ and $k\in K$, denoted $K\lhd G$.
Quotient group, $G/K$
Given a normal subgroup $K\lhd G$, the quotient group is comprised of the cosets of K, denoted $G/K$. By Lagrange's Theorem, the order of a quotient group is $|G/K|=|G|/|K|$.

## Key Examples

Cyclic group
A cyclic group is generated by a single element; all such groups are isomorphic to either the finite $\mathbb{Z}_m$ (integers modulo m) or the integers $\mathbb{Z}$.
Group of Units, $U(n)$
The set of elements of $\mathbb{Z}_n$ which are relatively prime to n form the group of units $U(n)$ under multiplication.
Symmetric group, Sn
The symmetric group Sn is that formed from the set of all permutations of n elements, with operation being the composition of permutations. The even permutations form the alternating group An.
Dihedral Group, Dn
The dihedral group Dn is the group of symmetries of an n-polygon, consisting of n rotations and n reflections.
Matrix Group
The set of matrices $GL(n,\mathbb{R})$ or $GL(n,\mathbb{C})$, with nonzero determinant and entries in $\mathbb{R}$ or $\mathbb{C}$, form a group under matrix multiplication.

## Key Theorems

Cayley's Theorem
Every group G is isomorphic to a subgroup of a symmetric group: either Sn where $n=|G|$ if its order is finite, or $S_G$ (the permutations on elements of G) if its order is infinite. This result is most useful in the classification of groups.
First Isomorphism Theorem
Given a homomorphism $f:G\to H$, the kernel $\ker(f)=\{g\in G:f(g)=e\}$ is a normal subgroup, and $G/\ker(f)\cong f(G)$. Alternately, if $K=\ker{f}$ and $\pi:G\to G/K$ is defined properly, then there exists an isomorphism $\phi$ with $f=\phi\circ\pi$. The next two results are consequences of this theorem.
Second Isomorphism Theorem
If H and K are subgroups of G, with $H\lhd G$ a normal subgroup, then the set of products $HK$ forms a group, $H\cap K$ is a normal subgroup of $K$, and $K/(H\cap K)\cong HK/H$.
Third Isomorphism Theorem
If H and K are both normal subgroups of G, and $K<H$, then $H/K \lhd G/K$ and $(G/K)/(H/K)\cong G/H$.

## Special Subgroups and Factor Groups

Automorphism Group, $\mathrm{Aut}(G)$
The set of automorphisms of G forms a group.
Inner Automorphism Group, $\mathrm{Inn}(G)$
Automorphisms of the form $\phi_a(x)=axa^{-1}$, for $a\in G$, are called inner automorphisms and form a normal subgroup of $\mathrm{Aut}(G)$.
Outer Automorphism Group, $\mathrm{Out}(G)$
The quotient group $\mathrm{Aut}(G) / \mathrm{Inn}(G)$ is the outer automorphism group.
Cyclic Subgroup, $\langle a\rangle$
For any $a\in G$, the elements $\langle a\rangle \equiv \{1\}\cup\{a,a^2,\ldots\}\cup\{a^{-1},a^{-2},\ldots\}$ form the cyclic subgroup generated by a.
Center, $Z(G)$
The center $Z(G)$ of a group G consists of those elements that commute with all others. (The "Z" is from the German Zentrum, which means "center".) Elements of $Z(G)$ are said to be central elements of the group. $Z(G)$ is a normal subgroup, and $\frac{G}{Z(G)} \cong \mathrm{Inn}(G)$. One has $Z(G)=G$ if and only if G is abelian.
G/Z Theorem
If the quotient group $\frac{G}{Z(G)}$ is cyclic, then G is abelian.
Centralizer, $C(a)$
For any element $a\in G$, the centralizer $C(a)$ of a is the elements which commute with it: $C(a)=\{g\in G: ga=ag\}$$]]. One can extend this definition to centralizers$C(H)$of subgroups$H < G$. Normalizer,$N(H)$For any subgroup$H<G$, the set of elements x with$xH = Hx$is called the normalizer$N(H)$of H. N/C Theorem$C(H)$is a normal subgroup of$N(H)$, and the factor group$\frac{N(H)}{C(H)}$is isomorphic to a subgroup of$\mathrm{Aut}(H)$. ## Group Actions Group action The group G acts on a set X if there is a function$G\times X\to X$such that$(g,x)\mapsto g\cdot x$with$e\cdot x=x$for all x and$(gh)\cdot x = g\cdot(h\cdot x)$. A typical example is a permutation group acting on the elements of a set. Right action The left action is defined under "group action" above. The right action is a less commonly used variant defined by$(g,x)\mapsto x\cdot g$with$x\cdot e=x$and$x\cdot(gh) = (x\cdot g)\cdot h$. Conjugacy action The left action$(g,x)\mapsto gxg^{-1}$is the conjugacy action of G on itself. The variant$(g,x)\mapsto g^{-1}xg$is a right action. Orbit,$\mathrm{orb}(x)$or$Gx$Given an action of G on a set X, the orbit of$x\in X$is the set of possible image points of x under the action:$Gx = \{g\cdot x : g\in G\}$. Stabilizer,$\mathrm{stab}(x)$or$G_x$Given an action of G on a set X, the stabilizer of$x\in X$is the set of elements which fix x under the action:$G_x = \{g \in G : g \cdot x = x\}$. Orbit-Stabilizer Theorem There is a bijection between the space of cosets$G / G_x$and the orbit$Gx$. If G and X are finite, then$|G| = |G_x| \: |Gx|$. ## Solvability Simple group A simple group is one with no nontrivial normal subgroups. Normal series A normal series is a sequence of normally nested groups$H_0\lhd H_1\lhd \cdots \lhd H_k$. This construction leads to a sequence of factor groups$\frac{H_1}{H_0}, \ldots, \frac{H_k}{H_{k-1}}$called composition factors. Composition series A composition series is a normal series in which all nontrivial factor groups are simple. Jordan-Holder theorem Any two composition series of a group are equivalent; thus, the length (of series ending in the identity) is an invariant of the group. Solvable group A group is solvable if there is a normal series$\{e\}=H_0\lhd H_1\lhd \cdots \lhd H_k$for which all factor groups$\frac{H_{i+1}}{H_i}$are abelian. For$n\ge5$, the symmetric group Sn is not solvable. ## Sylow Theory Conjugacy class,$\mathrm{cl}(a)$The conjugacy class of an element$a\in G$is the set$\mathrm{cl}(a)=\{gag^{-1} : g \in G$. Conjugacy classes partition G and are rarely subgroups. The number of elements in$\mathsf{cl}(a)$is the index$|G:C(a)|$. Class Formula The order of G is$\sum|G:C(a)|$, the sum taken over conjugacy classes. Alternately,$|G|=|Z(G)|+\sum|G:C(a)|$. Sylow's First Theorem If$p^k$divides$|G|$, then G has a subgroup of order$p^k$. The maximal such subgroup is the Sylow p-subgroup. Sylow's Second Theorem Every subgroup$H<G$with order$|H|=p^k$is contained in some Sylow p-subgroup. Sylow's Third Theorem Any two Sylow p-subgroups are conjugate; the number np of such is equal to 1, modulo p, and also divides$|G|$. The subgroup is unique if and only if it is normal. Solvability of p-groups Every finite p-group is solvable, since it has a nontrivial center. ## Free Groups Free group Given a set of letters S, the free group on S is generated by formal finite sequences (called words) of elements in S and their inverses. Universal mapping property Every group is a homomorphic image of a free group. Generator and relation expression of groups Every group is a factor group of a free group, hence is definable by a set of generators and relations. Free abelian group On a set of generators$\{x_i\}$, the free abelian group is$\langle x_1\rangle\oplus\cdots\oplus\langle x_k\rangle$. Torsion subgroup The torsion subgroup of an abelian group consists of all finite-order elements. Internal direct sums in abelian groups A group G may be written as a direct product$G=H\times K$if$G=HK$for normal subgroups H and K with$H\cap K=\{e\}$. In this case,$H\oplus K\cong H\times K$, so the internal product is isomorphic to the external product. Decomposition of abelian torsion groups The torsion subgroup of an abelian group is isomorphic to$\mathbb{Z}_{p_1}^{n_1}\oplus\cdots\oplus\mathbb{Z}_{p^k}^{n_k}$where the primes pi and the coefficients ni are unique up to reordering. The remainder of the group will then be$\mathbb{Z}^{n_{k+1}}$. # Rings See the article on beginning abstract algebra for more details. ## Basics Ring A ring is a set R with operations + and * such that$(R,+)$is an abelian group, R is associative under$\cdot$, and R is distributive, meaning$a(b+c)=ab+ac)$. The additive identity is usually denoted 0; the multiplicative identity, if it exists, is usually denoted 1. Commutative Ring In a commutative ring, multiplication is commutative. Ring with unity A ring with unity has a multiplicative identity 1. Field A field is a ring R in which both operations are abelian groups (in particular,$(R,*)$is). Left Ideal A left ideal is a subring$A\subset R$with$ar\in A$for all$a\in A,r\in R$. A right ideal is similarly defined. Ideal or 2-Sided Ideal A 2-sided ideal is a subring$A\subset R$which is both a left and right ideal. Quotient Ring,$R/A$Given an ideal$A\subset R$, a quotient ring is the set of (additive) cosets of A, denoted$R/A$. Isomorphism Theorems There are three isomorphism theorems for rings, analogous to those for groups. ## Key Examples Polynomial Ring,$R[x]$A polynomial ring R is the formal set$R[x]$consisting of$a_0+a_1x+\cdots+a_nx^n$with$a_i\in R$. If R is an integral domain, so is$R[x]$. If R is a UFD, so is$R[x]$. If R is a field,$R[x]$is a PID. Matrix Ring,$M_n(R)$Given a ring R, one can form the ring$M_n(R)$of$n\times n$matrices with entries in R. The group operations are matrix addition and multiplication. Group Ring or Group Algebra,$R[G]$Given a ring R and a group G, the group ring is the set of formal sums$\sum_{g_i\in G} \alpha_i g_i$with$\alpha_i\in R$. Addition is defined in the obvious way, and multiplication is defined by$\alpha_i g_i \cdot \alpha_j g_j=\alpha_i \alpha_j g_i g_j$. This behaves much like a vector space over R equipped with a multiplication, whose basis consists of the elements of G (precisely, it is an R-module). ## Integral Domains Zero Divisor In a ring R, a zero divisor is a nontrivial element$a\in R$such that$a b=0$for some nonzero$b\in R$. Integral Domain An integral domain is a commutative ring with unity and no zero divisors. Equivalently, the ring satisfies cancellation laws$ab=ac \Rightarrow b=c$and$ba=ca \Rightarrow b=c$. Prime Ideal A prime ideal in a ring R is an ideal$A\subset R$such that$ab\in A$implying either$a\in A$or$b\in A$. In this case, the quotient ring$R/A$is an integral domain. Maximal Ideal A maximal ideal in a ring R is an ideal$A\subset R$contained in no other nontrivial ideal of R. In this case, the quotient ring$R/A$is a field. Chinese Remainder Theorem Given a ring R with pairwise coprime ideals$I_1,\ldots,I_n$and elements$a_1,\ldots,a_n\in R$, there exists a single$r\in R$such that$r+I_i=a_i+I_i$for all i. ## Factorization Principal Ideal A principal ideal in a commutative ring R is one of the form$\langle a\rangle=\{ra:r\in R\}$for some$a\in R$. It is the smallest subring containing a. Principal Ideal Domain (PID) A principal ideal domain is an integral domain in which every ideal is principal. Prime Element In an integral domain R, a prime element is an element$a\in R$for which$a|bc$implies either$a|b$or$a|c$. Irreducible Element In an integral domain R, an irreducible element is an element$a\in D$for which$a=bc$implies that either b or c is a unit. Unique Factorization Domain (UFD) A unique factorization domain is an integral domain in which the unique factorization property holds, meaning that every element can be factored into irreducible elements, and the factorization is unique up to units. Every PID is a UFD. Euclidean Ring/Domain (ED) A Euclidean domain is an integral domain R with a division algorithm. Thus, there is some function$d:R^*\to\mathbb{Z}_0^+$such that$d(a)\leq d(ab)$,$b\neq 0$implies there are$q,r\in R$such that$a=bq+r$and$d(r)<d(b)$. Every ED is a PID. ## Artinian/Noetherian Rings Simple Ring A simple ring is one with no nontrivial ideals. Semisimple Ring A semisimple ring may be expressed as a direct sum of minimal ideals (??? needs checking) Division Ring or Skew Field A division ring ring with every nonzero element having an inverse, but not necessarily commutative. Given a division ring D, the ring of n x n matrices over D is simple. Artinian Ring An Artinian ring is one in which every descending chain of ideals$I_1\supset I_2\supset\cdots$stops. Noetherian Ring A Noetherian ring is one in which every ascending chain of ideals$I_1\subset I_2\subset\cdots$stops. More simply, every ideal is finitely generated. Wedderburn's Theorem for Simple Artinian Rings A simple Artinian ring is isomorphic to a ring of matrices over a division ring. Hilbert Basis Theorem If R is a commutative Noetherian ring, then so is R[x]. ## Localization Multiplicative Set In a ring, a multiplicative set is a subset that includes the unity and is closed under multiplication. Fraction Field or Field of Quotients In an integral domain D, the field of quotients is the set of quotients of elements of D, modulo an appropriate equivalence relation. The default example is the rationals as constructed from the integers. Local Ring A local ring is one with a unique maximal ideal. The other elements are precisely the units of the ring. # Modules This part of the glossary is still rather incomplete!! ## Basics Module Given a ring R an R-module is an abelian group M with a scalar multiplication law$R\times M\to M$that satisfies the expected laws (i.e. similar laws to those for vector spaces). Examples of Modules Vector spaces over a field F are F-modules. Abelian groups are$\mathbb{Z}$-modules, looking at the exponents of group elements. A commutative ring R is an S-module for any subring$S\subset R$, including itself. Modules over Matrix Rings One can form a module over a matrix ring$M_n(R)$as the set of n-tuples of elements of R, using standard matrix multiplication. Modules over Group Rings One can form a module over a group ring$R[G]$, just as for any other ring. Exact Sequence A sequence$\cdots \to M_{n+1} \to M_n\to M_{n-1} \to \cdots$of R-modules and R-maps$f_i:M_i\to M_{i-1}$is exact if$\mathrm{image} f_{i+1}=\ker f_i$for all i. Missing Terms: Exactness Properties of$\mathsf{Hom}$## Free Modules Free Module A free R-module is one which is isomorphic to a direct sum of copies of R. Every R-module is a quotient of a free R-module. Invariance of Rank For a commutative ring R, every free R-module has the same rank, meaning the same number of elements in any basis. This is not generally true in non-commutative rings. Presentation of a Module Every R-module is a quotient of a free R-module. The free module provides the generators (basis elements) for the presentation, while the relations are provided by the factor submodule. Finitely Generated Module over a PID When R is a Principal Ideal Domain, a finitely generated R-module M is a direct sum$M=tM\oplus F$, where tM is the torsion part of the module and F is a free module. This implies primary decomposition theorems for both abelian groups and torsion R-modules. Missing Terms: Applications to Canonical Forms of Matrices and Abelian Groups ## Tensor Products Tensor Product The tensor product of R-modules A and B is the space$A\otimes_R B$and a map$f:A\times B \to A\otimes_R B$such that for all bilinear maps$g:A\times B\to G$into an abelian group G, there exists a map$h:A\otimes_R B\to G$which commutes with the previous two, i.e.$g = h\circ f$. One says that all bilinear maps factor through the map f. Missing Terms: Localization, Algebras and Base Change, Exactness Properties of Tensor Products, Exterior Algebra ## Projective/Injective Modules Projective Module A projective module P is one in which every short exact sequence$0\to A\to^i B\to^p P\to 0$is split. Equivalently, given a surjection$A\to B$and a map$P\to B$, there exists a pullback map$P\to A$commuting with the other two. Injective Module An injective module E is one in which every short exact sequence$0\to E\to^i A\to^p B\to 0$is split. Equivalently, given an injection$A\to B$and a map$E\to A$, there exists a map$E\to B$which commutes with the other two. Missing Terms: Homology, The Snake Lemma, Facts on derived functors including$\mathsf{Tor}$and$\mathsf{Ext}$# Field Theory See the article on beginning abstract algebra for more details. Field A field is a set F with operations + and * such that (i) F is a group under +, (ii) F, with the additive identity removed, is a group under *, and (iii) the operations satisfy the distributive law a*(b+c)=a*b+a*c. The multiplicative group is denoted$F^*$. Finite Field A finite field must have order pn for some prime p. It is sometimes denoted$GF(p^n)$. The field's multiplicative group$GF(p^n)^*$is cyclic, and the fields additive group is isomorphic to the direct sum$\mathbb{Z}/p\mathbb{Z}\oplus\cdots\oplus\mathbb{Z}/p\mathbb{Z}$. ## Field extensions Field Extension A field extension of a field F is a field E that contains F, written E > F. Algebraic Element An element a in an extension field E > F is algebraic over F if it is the zero of a polynomial in F[x]. An algebraic extension is one in which every element is algebraic. Transcendental Element A transcendental element a in an extension field E > F is one which is not algebraic. Characteristic The characteristic of a field is the order of the multiplicative unit, or 0 if that order is infinite. Algebraic Closure The algebraic closure of a field F is the smallest extension field containing all of its algebraic elements. One says that every polynomial over F splits (i.e. factors) in this extension. Transcendence Basis The transcendence basis of a field extension E > F is a maximal algebraically independent subset$B\subset E$, with transcendence degree defined to be |B|. Intuitively, it is the number of elements which must be adjoined to F to obtain E. ## Splitting fields/normal extensions Splitting Field Given a polynomial$p(x)\in F[x]$, a splitting field for p(x) is an extension field E > F that contains all the roots of the polynomial, allowing the polynomial to be split (i.e. factored). Normal Extension A normal extension field is the splitting field of some set of polynomials. Separable Polynomial A separable polynomial is one with no repeated roots. Separable Extension An extension E > F is separable if all of its elements are separable, meaning either the elements are transcendental or their minimal polynomial is separable. Missing Terms: Extension of Isomorphisms ## Galois Theory Galois Extension An extension E > F is Galois if it is the splitting field of a polynomial in F[x]. These are separable extensions. Galois Group The Galois group$\mathsf{Gal}(E/F)$of a field extension E > F is the group of automorphisms of E that fix all elements of F. It is isomorphic to a subgroup of the symmetric group Sn, where n is the degree of the polynomial with splitting field E (since its elements must permute the roots of the polynomial). Galois Correspondence Given a Galois extension E > F, there is a one-to-one correspondence between intermediate fields K (with E > K > F) and subgroups of$\mathsf{Gal}(E/F)$, given by the map$K\mapsto\mathsf{Gal}(E/K)$. Cyclic Extensions In a Galois extension where E / F is of prime degree p, and F includes a primitive pth root of unity, i.e. root of the equation xp=1, then the Galois group is$\mathbb{Z}/p\mathbb{Z}$and$E=F(\beta)$for some$\beta^p \in F$. Roots of Unity If$\omega$is a primitive nth root of unity, then$\mathsf{Gal}(\mathbb{Q}(\omega)/\mathbb{Q})\cong\mathrm{U}(n)$. Ruler and Compass Constructions An n-gon is constructable if and only if a primitive nth root of unity is constructable. The only possible choices are$2^k\prod p_i$, where the pi are primes of the form 2m+1. Solvable by Radicals A polynomial$f(x)\in F[x]$is solvable by radicals if it splits in some extension$F(a_1,\ldots,a_n)$and there exist$k_i\in\mathbb{Z}^+$such that$a_i^{k_i}\in F(a_1,\ldots,a_{i-1})$. In this case, the Galois group of the extension$F(a_1,\ldots,a_n)$is solvable. Missing Terms: Norms, Traces, Computations of Galois Groups # Representation Theory of Groups See the article on group representations for more details. Representation (of a Group) A group representation is a homomorphism$\sigma:G\to GL(V)$, where GL(V) is the general linear group of invertible transformations on some vector space V. Typically, one takes V to be the reals R or the complex numbers C. Module Correspondence Modules can be understood as abelian groups with representations. Trivial Representation The trivial representation takes all of G to the identity transformation. Regular Representation The regular representation is formed by the group acting on itself to form a module. Faithful Representation A faithful representation maintains the full group structure. A brief way to state this condition is$\ker\sigma=0$. Equivalent Representations Representations$\sigma:G\to GL(V)$and$\tau:G\to GL(W)$are equivalent if there exists an isomorphism$\alpha:V\to W$such that$\tau(g)=\alpha\circ\sigma\circ\alpha^{-1}$. Equivalent representations come from the same module. Irreducible Representation An irreducible representation has no nontrivial invariant subspaces (called subrepresentations. Equivalently, there are no nontrivial submodules. Reducible Representation For certain V, reducible representations may be written in block-triangular form. Completely reducible representations may be written in block diagonal form, meaning they can be decomposed as a direct sum of other representations. Character The character of a representation$\sigma:G\to GL(V)$is its trace, the function$\chi_\sigma:G\to K$given by$\chi_\sigma(g) = \mathrm{tr}(\sigma(g))$. It is constant on conjugacy classes. Class function A class function is any function which is constant on conjugacy classes (such as the trace). The irreducible characters (those which come from irreducible representations) form a basis for the class functions of a group. # Categories See the article on category theory for more details. Category A category is a set of objects$\mathcal{C}$, a set of morphisms comprising Hom(A,B) between ordered pairs of objects$(A,B) \in \mathcal{S}\times\mathcal{C}$, and an associative morphism composition rule$\mathrm{Hom}(A,B)\times\mathrm{Hom}(B,C)\to\mathrm{Hom}(A,C)$. The sets of morphisms must also include an identity morphism$1_A\in\mathrm{Hom}(A)$for each$A\in\mathcal{C}$. Examples include sets with functions between sets, groups with homomorphisms, and commutative rings with ring homomorphisms. Equivalence morphism A morphism$f:A\to B$is an equivalence if there exists another morphism$g:B\to A$such that$g\circ f=1_A$and$f\circ g=1_B$. Examples include bijective maps between sets, group isomorphisms, and ring isomorphisms. Functor A functor is a map T between two categories that takes objects to objects, takes maps to maps, takes identity morphisms to identity morphisms, and preserves the composition rule. The$\mathrm{Hom}$functor Given an object A in a category$\mathcal{C}$, the$\mathrm{Hom}$functor is the functor mapping objects B to collections of morphisms$B\mapsto\mathrm{Hom}(A,B)$and morphisms$f\in\mathsf{Hom}(B,B')$to elements of$\mathrm{Hom}(\mathrm{Hom}(A,B), \mathrm{Hom}(A,B'))$. In particular, the morphism f maps to the morphism$g\mapsto g\circ f$. Coproduct or Direct sum or Free product The direct sum of objects, denoted$A_1\sqcup A_2$or$A_1\oplus A_2$, is defined by the properties of its injection morphisms$\alpha_i:A_i\to A_1\sqcup A_2$: for every morphism$f_i\in\mathrm{Hom}(A_i,X)$, there exists a unique morphism$\theta:A_1\sqcup A_2\to X$such that$\theta\alpha_i=f_i$. Products The product of objects is the object$P=A_1\sqcap A_2$, together with morphisms$p_i\in\mathsf{Hom}(P,A_i)$satisfying the following property: for every morphism$f_i\in\mathrm{Hom}(X,A_i)$, there exists a unique morphism$\theta:X\to P$such that$p_i\theta=f_i\$. This differs from the coproduct in that the "arrows are reversed". It may give the same object as the coproduct.
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