Isomorphism Theorem Article Index for
Isomorphism 		Website Links For
Isomorphism 		 

Information About

Isomorphism Theorem
  CATEGORIES ABOUT ISOMORPHISM THEOREM
   
mathematical theorems
   
isomorphism theorems
   
APPAREL
BABY
BEAUTY
BOOKS
CAR TOYS
CELL PHONES
DVD'S
ELECTRONICS
GOURMET FOOD
GROCERIES
HEALTH & PERSONAL
HOME & GARDEN
JEWELRY
MUSIC
MUSIC INSTRUMENTS
OFFICE PRODUCTS
SOFTWARE
SPORTING GOODS
TOOLS & HARDWARE
TOYS
VIDEO GAMES
SHOPPING HOME
MORE SHOPPING...




HISTORY


The isomorphism theorems were originally formulated by Emmy Noether in her paper ''Abstrakter Aufbau der Idealtheorie in algebraischen Zahl- und Funktionenkörpern'' which was published in 1927 in Mathematische Annalen .

Three years later, B.L. Van Der Waerden published his influential ''Algebra,'' the first Abstract Algebra textbook that took the now-traditional 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


First we state the isomorphism theorems for Groups , where they take a simpler form and state important properties of Quotient Group s (also called factor groups). All three involve " Modding Out " by a Normal Subgroup .


First isomorphism theorem


If ''G'' and ''H'' are groups and ''f'' is a Homomorphism from ''G'' to ''H'', then the Kernel ''K'' of ''f'' is a Normal Subgroup of ''G'', and the Quotient Group ''G''/''K'' is Isomorphic to the Image of ''f'' and the Image of ''f'' is a Subgroup of ''H''.

If
:G, H ext{ are groups}\;
:f: G o H, f ext{ is a homomorphism}\;
then
:\operatorname{Ker}(f) riangleleft G
:G/\operatorname{Ker}(f) \cong \operatorname{Im}(f)
:\operatorname{Im}(f) ext{ is a subgroup of } H


Second isomorphism theorem (also known as the third isomorphism theorem)


Let ''H'' and ''K'' be subgroups of the group ''G'', and assume ''H'' is a subgroup of the Normalizer of ''K''. Then the Join ''HK'' of ''H'' and ''K'' is a subgroup of G, ''K'' is a normal subgroup of ''HK'', ''H'' ∩''K'' is a normal subgroup of H, and ''HK''/''K'' is Isomorphic to ''H''/(''H'' ∩''K'').

If
:H,K ext{ are subgroups of group } G \,
:H ext{ is a subgroup of } \operatorname{N_G}(K)
then
:HK ext{ is a subgroup of } G \,
:K riangleleft HK
:H \cap K riangleleft H
:HK/K \cong H/(H \cap K)


Third isomorphism theorem (also known as the second isomorphism theorem)


If ''M'' and ''N'' are normal subgroups of ''G'' such that ''M'' is contained in ''N'', then ''M'' is a normal subgroup of ''N'', ''N''/''M'' is a normal subgroup of ''G''/''M'', and (''G''/''M'')/(''N''/''M'') is Isomorphic to ''G''/''N''.

If
:M,N riangleleft G
:M \subseteq N
then
:M riangleleft N
:N/M riangleleft G/M
:(G/M)/(N/M) \cong G/N


RINGS AND MODULES


The isomorphism theorems are also valid for Modules over a fixed Ring ''R'' (and therefore also for Vector Space s over a fixed Field ). One has to replace the term "group" by "''R''-module", "subgroup" and "normal subgroup" by " Submodule ", and "factor group" by " Factor Module ".

For vector spaces, the first isomorphism theorem goes by the name of Rank-nullity Theorem .

The isomorphism theorems are also valid for rings, ring homomorphisms and Ideal s. One has to replace the term "group" by "ring", "subgroup" by "subring" and "normal subgroup" by "ideal", and "factor group" by " Factor Ring ".

The notation for the Join in both these cases is "''H'' + ''K''" instead of "''HK''".

We also need to mention the isomorphism theorems for topological vector spaces, Banach algebras etc.



GENERAL


To generalise this to universal algebra, normal subgroups need to be replaced by Congruence s.

Briefly, if ''A'' is an Algebra , a congruence on ''A'' is an equivalence relation \Phi on ''A'' which is a subalgebra when considered as a subset of ''A x A'' (the latter with the coordinate-wise operation structure). One can make the set of equivalence classes ''A/''\Phi into an algebra of the same type by defining the operations via representatives; this will be well-defined since \Phi is a subalgebra of ''A x A''.


First Isomorphism Theorem

If ''A'' and ''B'' are algebras, and ''f'' is a Homomorphism from ''A'' to ''B'', then the equivalence relation \Phi on ''A'' defined by ''a~b'' if and only if ''f(a)=f(b)'' is a congruence on ''A'', and the algebra ''A/''\Phi is isomorphic to the image of ''f'', which is a subalgebra of ''B''.


Second Isomorphism Theorem


Given an algebra ''A'', a subalgebra ''B'' of ''A'', and a congruence \Phi on ''A'', we let be the subset of ''A/''\Phi determined by all congruence classes that contain an element of ''B'', and we let \Phi_B be the intersection of \Phi (considered as a subset of ''A x A'') with ''B x B''. Then [B \Phi is a subalgebra of ''A/''\Phi, \Phi_B is a congruence on ''B'', and the algebra [B]\Phi is isomorphic to the algebra ''B/''\Phi_B.


Third Isomorphism Theorem


Let ''A'' be an algebra, and let \Phi and \Psi be two congruence relations on ''A'', with \Psi contained in \Phi. Then \Phi determines a congruence \Theta on ''A/''\Psi defined by '' if and only if ''a'' and ''b'' are equivalent modulo \Phi (where ''[a '' represents the \Psi-equivalence class of ''a''), and ''A/''\Phi is isomorphic to ''(A/''\Psi'')/''\Theta.


REFERENCES




SEE ALSO




EXTERNAL LINKS

  • .

  • .

  • .