Isomorphism theorem

In mathematics, specifically abstract algebra, the isomorphism theorems (also known as Noether's isomorphism theorems) are three theorems that describe the relationship between quotients, homomorphisms, and subobjects. Versions of the theorems exist for groups, rings, vector spaces, modules, Lie algebras, and various other algebraic structures. In universal algebra, the isomorphism theorems can be generalized to the context of algebras and congruences. P G L 2 ( C ) := G L 2 ( C ) / ( C × I ) ≅ S L 2 ( C ) / { ± I } =: P S L 2 ( C ) {displaystyle PGL_{2}(mathbb {C} ):=GL_{2}(mathbb {C} )/(mathbb {C} ^{ imes }!I)cong SL_{2}(mathbb {C} )/{pm I}=:PSL_{2}(mathbb {C} )} In mathematics, specifically abstract algebra, the isomorphism theorems (also known as Noether's isomorphism theorems) are three theorems that describe the relationship between quotients, homomorphisms, and subobjects. Versions of the theorems exist for groups, rings, vector spaces, modules, Lie algebras, and various other algebraic structures. In universal algebra, the isomorphism theorems can be generalized to the context of algebras and congruences. 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 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. The first instance of the isomorphism theorems that we present occurs in the category of abstract groups. Note that some sources switch the numbering of the second and third theorems. Another variation encountered in the literature, particularly in Van der Waerden's Algebra and Pinter's A Book of Abstract Algebra, is to call first isomorphism theorem the Fundamental Homomorphism Theorem and consequently to decrement the numbering of the remaining isomorphism theorems by one. Finally, in the most extensive numbering scheme, the lattice theorem (also known as the correspondence theorem) is sometimes referred to as the fourth isomorphism theorem. Let G and H be groups, and let φ: G → H be a homomorphism. Then: In particular, if φ is surjective then H is isomorphic to G / ker(φ). Let G {displaystyle G} be a group. Let S {displaystyle S} be a subgroup of G {displaystyle G} , and let N {displaystyle N} be a normal subgroup of G {displaystyle G} . Then the following hold: Technically, it is not necessary for N {displaystyle N} to be a normal subgroup, as long as S {displaystyle S} is a subgroup of the normalizer of N {displaystyle N} in G {displaystyle G} . In this case, the intersection S ∩ N {displaystyle Scap N} is not a normal subgroup of G {displaystyle G} , but it is still a normal subgroup of S {displaystyle S} . This isomorphism theorem has been called the 'diamond theorem' due to the shape of the resulting subgroup lattice with S N {displaystyle SN} at the top, S ∩ N {displaystyle Scap N} at the bottom and with N {displaystyle N} and S {displaystyle S} to the sides. It has even been called the 'parallelogram theorem' because in the resulting subgroup lattice the two sides assumed to represent the quotient groups ( S N ) / N {displaystyle (SN)/N} and S / ( S ∩ N ) {displaystyle S/(Scap N)} are 'equal' in the sense of isomorphism.