# Isomorphism theorems

Contents 1 History 2 Groups 2.1 Note on numbers and names 2.2 Statement of the theorems 2.2.1 Theorem A (groups) 2.2.2 Theorem B (groups) 2.2.3 Theorem C (groups) 2.2.4 Theorem D (groups) 2.3 Discussion 3 Rings 3.1 Theorem A (rings) 3.2 Theorem B (rings) 3.3 Theorem C (rings) 3.4 Theorem D (rings) 4 Modules 4.1 Theorem A (modules) 4.2 Theorem B (modules) 4.3 Theorem C (modules) 4.4 Theorem D (modules) 5 Universal algebra 5.1 Theorem A (universal algebra) 5.2 Theorem B (universal algebra) 5.3 Theorem C (universal algebra) 5.4 Theorem D (universal algebra) 6 Note 7 References History The isomorphism theorems were formulated in some generality for homomorphisms of modules by Emmy Noether in her paper Abstrakter Aufbau der Idealtheorie in algebraischen Zahl- und Funktionenkörpern, which was published in 1927 in Mathematische Annalen. Less general versions of these theorems can be found in work of Richard Dedekind and previous papers by Noether.

Three years later, B.L. van der Waerden published his influential Moderne Algebra the first abstract algebra textbook that took the groups-rings-fields approach to the subject. Van der Waerden credited lectures by Noether on group theory and Emil Artin on algebra, as well as a seminar conducted by Artin, Wilhelm Blaschke, Otto Schreier, and van der Waerden himself on ideals as the main references. The three isomorphism theorems, called homomorphism theorem, and two laws of isomorphism when applied to groups, appear explicitly.

Groups We first present the isomorphism theorems of the groups.

Note on numbers and names Below we present four theorems, labelled A, B, C and D. They are often numbered as "First isomorphism theorem", "Second..." and so on; however, there is no universal agreement on the numbering. Here we give some examples of the group isomorphism theorems in the literature. Notice that these theorems have analogs for rings and modules.

Comparison of the names of the group isomorphism theorems Comment Author Theorem A Theorem B Theorem C No "third" theorem Jacobson[1] Fundamental theorem of homomorphisms (Second isomorphism theorem) "often called the first isomorphism theorem" van der Waerden,[2] Durbin[4] Fundamental theorem of homomorphisms First isomorphism theorem Second isomorphism theorem Knapp[5] (No name) Second isomorphism theorem First isomorphism theorem Grillet[6] Homomorphism theorem Second isomorphism theorem First isomorphism theorem Three numbered theorems (Other convention per Grillet) First isomorphism theorem Third isomorphism theorem Second isomorphism theorem Rotman[7] First isomorphism theorem Second isomorphism theorem Third isomorphism theorem Fraleigh[8] (No name) Second isomorphism theorem Third isomorphism theorem Dummit & Foote[9] First isomorphism theorem Second or Diamond isomorphism theorem Third isomorphism theorem No numbering Milne[10] Homomorphism theorem Isomorphism theorem Correspondence theorem Scott[11] Homomorphism theorem Isomorphism theorem Freshman theorem It is less common to include the Theorem D, usually known as the lattice theorem or the correspondence theorem, as one of isomorphism theorems, but when included, it is the last one.

Statement of the theorems Theorem A (groups) See also: Fundamental theorem on homomorphisms Diagram of the fundamental theorem on homomorphisms Let G and H be groups, and let f : G → H be a homomorphism. Then: The kernel of f is a normal subgroup of G, The image of f is a subgroup of H, and The image of f is isomorphic to the quotient group G / ker(f).

In particular, if f is surjective then H is isomorphic to G / ker(f).

Theorem B (groups) Diagram for theorem B3. The two quotient groups (dotted) are isomorphic.

Let {displaystyle G} be a group. Let {displaystyle S} be a subgroup of {displaystyle G} , and let {displaystyle N} be a normal subgroup of {displaystyle G} . Then the following hold: The product {displaystyle SN} is a subgroup of {displaystyle G} , The intersection {displaystyle Scap N} is a normal subgroup of {displaystyle S} , and The quotient groups {displaystyle (SN)/N} and {displaystyle S/(Scap N)} are isomorphic.

Technically, it is not necessary for {displaystyle N} to be a normal subgroup, as long as {displaystyle S} is a subgroup of the normalizer of {displaystyle N} in {displaystyle G} . In this case, the intersection {displaystyle Scap N} is not a normal subgroup of {displaystyle G} , but it is still a normal subgroup of {displaystyle S} .

This theorem is sometimes called the isomorphism theorem,[10] diamond theorem[12] or the parallelogram theorem.[13] An application of the second isomorphism theorem identifies projective linear groups: for example, the group on the complex projective line starts with setting {displaystyle G=operatorname {GL} _{2}(mathbb {C} )} , the group of invertible 2 × 2 complex matrices, {displaystyle S=operatorname {SL} _{2}(mathbb {C} )} , the subgroup of determinant 1 matrices, and {displaystyle N} the normal subgroup of scalar matrices {displaystyle mathbb {C} ^{times }!I=left{left({begin{smallmatrix}a&0\0&aend{smallmatrix}}right):ain mathbb {C} ^{times }right}} , we have {displaystyle Scap N={pm I}} , where {displaystyle I} is the identity matrix, and {displaystyle SN=operatorname {GL} _{2}(mathbb {C} )} . Then the second isomorphism theorem states that: {displaystyle operatorname {PGL} _{2}(mathbb {C} ):=operatorname {GL} _{2}left(mathbb {C} )/(mathbb {C} ^{times }!Iright)cong operatorname {SL} _{2}(mathbb {C} )/{pm I}=:operatorname {PSL} _{2}(mathbb {C} )} Theorem C (groups) Let {displaystyle G} be a group, and {displaystyle N} a normal subgroup of {displaystyle G} . Then If {displaystyle K} is a subgroup of {displaystyle G} such that {displaystyle Nsubseteq Ksubseteq G} , then {displaystyle G/N} has a subgroup isomorphic to {displaystyle K/N} . Every subgroup of {displaystyle G/N} is of the form {displaystyle K/N} for some subgroup {displaystyle K} of {displaystyle G} such that {displaystyle Nsubseteq Ksubseteq G} . If {displaystyle K} is a normal subgroup of {displaystyle G} such that {displaystyle Nsubseteq Ksubseteq G} , then {displaystyle G/N} has a normal subgroup isomorphic to {displaystyle K/N} . Every normal subgroup of {displaystyle G/N} is of the form {displaystyle K/N} for some normal subgroup {displaystyle K} of {displaystyle G} such that {displaystyle Nsubseteq Ksubseteq G} . If {displaystyle K} is a normal subgroup of {displaystyle G} such that {displaystyle Nsubseteq Ksubseteq G} , then the quotient group {displaystyle (G/N)/(K/N)} is isomorphic to {displaystyle G/K} . Theorem D (groups) Main article: Lattice theorem The correspondence theorem (also known as the lattice theorem) is sometimes called the third or fourth isomorphism theorem.

The Zassenhaus lemma (also known as the butterfly lemma) is sometimes called the fourth isomorphism theorem.[14] Discussion The first isomorphism theorem can be expressed in category theoretical language by saying that the category of groups is (normal epi, mono)-factorizable; in other words, the normal epimorphisms and the monomorphisms form a factorization system for the category. This is captured in the commutative diagram in the margin, which shows the objects and morphisms whose existence can be deduced from the morphism {displaystyle f:Grightarrow H} . The diagram shows that every morphism in the category of groups has a kernel in the category theoretical sense; the arbitrary morphism f factors into {displaystyle iota circ pi } , where ι is a monomorphism and π is an epimorphism (in a conormal category, all epimorphisms are normal). This is represented in the diagram by an object {displaystyle ker f} and a monomorphism {displaystyle kappa :ker frightarrow G} (kernels are always monomorphisms), which complete the short exact sequence running from the lower left to the upper right of the diagram. The use of the exact sequence convention saves us from having to draw the zero morphisms from {displaystyle ker f} to {displaystyle H} and {displaystyle G/ker f} .

If the sequence is right split (i.e., there is a morphism σ that maps {displaystyle G/operatorname {ker} f} to a π-preimage of itself), then G is the semidirect product of the normal subgroup {displaystyle operatorname {im} kappa } and the subgroup {displaystyle operatorname {im} sigma } . If it is left split (i.e., there exists some {displaystyle rho :Grightarrow operatorname {ker} f} such that {displaystyle rho circ kappa =operatorname {id} _{{text{ker}}f}} ), then it must also be right split, and {displaystyle operatorname {im} kappa times operatorname {im} sigma } is a direct product decomposition of G. In general, the existence of a right split does not imply the existence of a left split; but in an abelian category (such as that of abelian groups), left splits and right splits are equivalent by the splitting lemma, and a right split is sufficient to produce a direct sum decomposition {displaystyle operatorname {im} kappa oplus operatorname {im} sigma } . In an abelian category, all monomorphisms are also normal, and the diagram may be extended by a second short exact sequence {displaystyle 0rightarrow G/operatorname {ker} frightarrow Hrightarrow operatorname {coker} frightarrow 0} .

In the second isomorphism theorem, the product SN is the join of S and N in the lattice of subgroups of G, while the intersection S ∩ N is the meet.

The third isomorphism theorem is generalized by the nine lemma to abelian categories and more general maps between objects.

Rings The statements of the theorems for rings are similar, with the notion of a normal subgroup replaced by the notion of an ideal.

Theorem A (rings) Let R and S be rings, and let φ : R → S be a ring homomorphism. Then: The kernel of φ is an ideal of R, The image of φ is a subring of S, and The image of φ is isomorphic to the quotient ring R / ker(φ).

In particular, if φ is surjective then S is isomorphic to R / ker(φ).

Theorem B (rings) Let R be a ring. Let S be a subring of R, and let I be an ideal of R. Then: The sum S + I = {s + i | s ∈ S, i ∈ I } is a subring of R, The intersection S ∩ I is an ideal of S, and The quotient rings (S + I) / I and S / (S ∩ I) are isomorphic. Theorem C (rings) Let R be a ring, and I an ideal of R. Then If {displaystyle A} is a subring of {displaystyle R} such that {displaystyle Isubseteq Asubseteq R} , then {displaystyle A/I} is a subring of {displaystyle R/I} . Every subring of {displaystyle R/I} is of the form {displaystyle A/I} for some subring {displaystyle A} of {displaystyle R} such that {displaystyle Isubseteq Asubseteq R} . If {displaystyle J} is an ideal of {displaystyle R} such that {displaystyle Isubseteq Jsubseteq R} , then {displaystyle J/I} is an ideal of {displaystyle R/I} . Every ideal of {displaystyle R/I} is of the form {displaystyle J/I} for some ideal {displaystyle J} of {displaystyle R} such that {displaystyle Isubseteq Jsubseteq R} . If {displaystyle J} is an ideal of {displaystyle R} such that {displaystyle Isubseteq Jsubseteq R} , then the quotient ring {displaystyle (R/I)/(J/I)} is isomorphic to {displaystyle R/J} . Theorem D (rings) Let {displaystyle I} be an ideal of {displaystyle R} . The correspondence {displaystyle Aleftrightarrow A/I} is an inclusion-preserving bijection between the set of subrings {displaystyle A} of {displaystyle R} that contain {displaystyle I} and the set of subrings of {displaystyle R/I} . Furthermore, {displaystyle A} (a subring containing {displaystyle I} ) is an ideal of {displaystyle R} if and only if {displaystyle A/I} is an ideal of {displaystyle R/I} .[15] Modules The statements of the isomorphism theorems for modules are particularly simple, since it is possible to form a quotient module from any submodule. The isomorphism theorems for vector spaces (modules over a field) and abelian groups (modules over {displaystyle mathbb {Z} } ) are special cases of these. For finite-dimensional vector spaces, all of these theorems follow from the rank–nullity theorem.

In the following, "module" will mean "R-module" for some fixed ring R.

Theorem A (modules) Let M and N be modules, and let φ : M → N be a module homomorphism. Then: The kernel of φ is a submodule of M, The image of φ is a submodule of N, and The image of φ is isomorphic to the quotient module M / ker(φ).

In particular, if φ is surjective then N is isomorphic to M / ker(φ).

Theorem B (modules) Let M be a module, and let S and T be submodules of M. Then: The sum S + T = {s + t | s ∈ S, t ∈ T} is a submodule of M, The intersection S ∩ T is a submodule of M, and The quotient modules (S + T) / T and S / (S ∩ T) are isomorphic. Theorem C (modules) Let M be a module, T a submodule of M.

If {displaystyle S} is a submodule of {displaystyle M} such that {displaystyle Tsubseteq Ssubseteq M} , then {displaystyle S/T} is a submodule of {displaystyle M/T} . Every submodule of {displaystyle M/T} is of the form {displaystyle S/T} for some submodule {displaystyle S} of {displaystyle M} such that {displaystyle Tsubseteq Ssubseteq M} . If {displaystyle S} is a submodule of {displaystyle M} such that {displaystyle Tsubseteq Ssubseteq M} , then the quotient module {displaystyle (M/T)/(S/T)} is isomorphic to {displaystyle M/S} . Theorem D (modules) Let {displaystyle M} be a module, {displaystyle N} a submodule of {displaystyle M} . There is a bijection between the submodules of {displaystyle M} that contain {displaystyle N} and the submodules of {displaystyle M/N} . The correspondence is given by {displaystyle Aleftrightarrow A/N} for all {displaystyle Asupseteq N} . This correspondence commutes with the processes of taking sums and intersections (i.e., is a lattice isomorphism between the lattice of submodules of {displaystyle M/N} and the lattice of submodules of {displaystyle M} that contain {displaystyle N} ).[16] Universal algebra To generalise this to universal algebra, normal subgroups need to be replaced by congruence relations.

A congruence on an algebra {displaystyle A} is an equivalence relation {displaystyle Phi subseteq Atimes A} that forms a subalgebra of {displaystyle Atimes A} considered as an algebra with componentwise operations. One can make the set of equivalence classes {displaystyle A/Phi } into an algebra of the same type by defining the operations via representatives; this will be well-defined since {displaystyle Phi } is a subalgebra of {displaystyle Atimes A} . The resulting structure is the quotient algebra.

Theorem A (universal algebra) Let {displaystyle f:Arightarrow B} be an algebra homomorphism. Then the image of {displaystyle f} is a subalgebra of {displaystyle B} , the relation given by {displaystyle Phi :f(x)=f(y)} (i.e. the kernel of {displaystyle f} ) is a congruence on {displaystyle A} , and the algebras {displaystyle A/Phi } and {displaystyle operatorname {im} f} are isomorphic. (Note that in the case of a group, {displaystyle f(x)=f(y)} iff {displaystyle f(xy^{-1})=1} , so one recovers the notion of kernel used in group theory in this case.) Theorem B (universal algebra) Given an algebra {displaystyle A} , a subalgebra {displaystyle B} of {displaystyle A} , and a congruence {displaystyle Phi } on {displaystyle A} , let {displaystyle Phi _{B}=Phi cap (Btimes B)} be the trace of {displaystyle Phi } in {displaystyle B} and {displaystyle [B]^{Phi }={Kin A/Phi :Kcap Bneq emptyset }} the collection of equivalence classes that intersect {displaystyle B} . Then {displaystyle Phi _{B}} is a congruence on {displaystyle B} , {displaystyle [B]^{Phi }} is a subalgebra of {displaystyle A/Phi } , and the algebra {displaystyle [B]^{Phi }} is isomorphic to the algebra {displaystyle B/Phi _{B}} . Theorem C (universal algebra) Let {displaystyle A} be an algebra and {displaystyle Phi ,Psi } two congruence relations on {displaystyle A} such that {displaystyle Psi subseteq Phi } . Then {displaystyle Phi /Psi ={([a']_{Psi },[a'']_{Psi }):(a',a'')in Phi }=[ ]_{Psi }circ Phi circ [ ]_{Psi }^{-1}} is a congruence on {displaystyle A/Psi } , and {displaystyle A/Phi } is isomorphic to {displaystyle (A/Psi )/(Phi /Psi ).} Theorem D (universal algebra) Let {displaystyle A} be an algebra and denote {displaystyle operatorname {Con} A} the set of all congruences on {displaystyle A} . The set {displaystyle operatorname {Con} A} is a complete lattice ordered by inclusion.[17] If {displaystyle Phi in operatorname {Con} A} is a congruence and we denote by {displaystyle left[Phi ,Atimes Aright]subseteq operatorname {Con} A} the set of all congruences that contain {displaystyle Phi } (i.e. {displaystyle left[Phi ,Atimes Aright]} is a principal filter in {displaystyle operatorname {Con} A} , moreover it is a sublattice), then the map {displaystyle alpha :left[Phi ,Atimes Aright]to operatorname {Con} (A/Phi ),Psi mapsto Psi /Phi } is a lattice isomorphism.[18][19] Note ^ Jacobson (2009), sec 1.10 ^ van der Waerden, Algebra (1994). ^ Durbin (2009), sec. 54 ^ [the names are] essentially the same as [van der Waerden 1994][3] ^ Knapp (2016), sec IV 2 ^ Grillet (2007), sec. I 5 ^ Rotman (2003), sec. 2.6 ^ Fraleigh (2003), Chap. 34 ^ Dummit, David Steven (2004). Abstract algebra. Richard M. Foote (Third ed.). Hoboken, NJ. pp. 97–98. ISBN 0-471-43334-9. OCLC 52559229. ^ Jump up to: a b Milne (2013), Chap. 1, sec. Theorems concerning homomorphisms ^ Scott (1964), secs 2.2 and 2.3 ^ I. Martin Isaacs (1994). Algebra: A Graduate Course. American Mathematical Soc. p. 33. ISBN 978-0-8218-4799-2. ^ Paul Moritz Cohn (2000). Classic Algebra. Wiley. p. 245. ISBN 978-0-471-87731-8. ^ Wilson, Robert A. (2009). The Finite Simple Groups. Graduate Texts in Mathematics 251. Springer-Verlag London. p. 7. doi:10.1007/978-1-84800-988-2. ISBN 978-1-4471-2527-3. ^ Dummit, David S.; Foote, Richard M. (2004). Abstract algebra. Hoboken, NJ: Wiley. p. 246. ISBN 978-0-471-43334-7. ^ Dummit and Foote (2004), p. 349 ^ Stanley and Sankappanavar (2012), p. 37 ^ Stanley and Sankappanavar (2012), p. 49 ^ William Sun, (https://math.stackexchange.com/users/413924/william-sun). "Is there a general form of the correspondence theorem?". Mathematics StackExchange. Retrieved 20 July 2019. {{cite web}}: External link in |first1= (help) References Emmy Noether, Abstrakter Aufbau der Idealtheorie in algebraischen Zahl- und Funktionenkörpern, Mathematische Annalen 96 (1927) pp. 26–61 Colin McLarty, "Emmy Noether's 'Set Theoretic' Topology: From Dedekind to the rise of functors". The Architecture of Modern Mathematics: Essays in history and philosophy (edited by Jeremy Gray and José Ferreirós), Oxford University Press (2006) pp. 211–35. Jacobson, Nathan (2009), Basic algebra, vol. 1 (2nd ed.), Dover, ISBN 9780486471891 Paul M. Cohn, Universal algebra, Chapter II.3 p. 57 Milne, James S. (2013), Group Theory, 3.13 van der Waerden, B. I. (1994), Algebra, vol. 1 (9 ed.), Springer-Verlag Dummit, David S.; Foote, Richard M. (2004). Abstract algebra. Hoboken, NJ: Wiley. ISBN 978-0-471-43334-7. Burris, Stanley; Sankappanavar, H. P. (2012). A Course in Universal Algebra (PDF). ISBN 978-0-9880552-0-9. W. R. Scott (1964), Group Theory, Prentice Hall John R. Durbin (2009). Modern Algebra: An Introduction (6 ed.). Wiley. ISBN 978-0-470-38443-5. Anthony W. Knapp (2016), Basic Algebra (Digital second ed.) Pierre Antoine Grillet (2007), Abstract Algebra (2 ed.), Springer Joseph J. Rotman (2003), Advanced Modern Algebra (2 ed.), Prentice Hall, ISBN 0130878685 Categories: Isomorphism theorems

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