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In mathematics, specifically abstract algebra, the isomorphism theorems (also known as Noether's isomorphism theorems) are 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. 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. Theorem A (groups) 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). This theorem is usually called the first isomorphism theorem. Theorem B (groups) 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: The product SN{\displaystyle SN} is a subgroup of G{\displaystyle G}, The subgroup N{\displaystyle N} is a normal subgroup of SN{\displaystyle SN}, The intersection S∩N{\displaystyle S\cap N} is a normal subgroup of S{\displaystyle S}, and The quotient groups (SN)/N{\displaystyle (SN)/N} and S/(S∩N){\displaystyle S/(S\cap N)} are isomorphic.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, N{\displaystyle N} is not a normal subgroup of G{\displaystyle G}, but N{\displaystyle N} is still a normal subgroup of the product SN{\displaystyle SN}. This theorem is sometimes called the second isomorphism theorem, diamond theorem or the parallelogram theorem.An application of the second isomorphism theorem identifies projective linear groups: for example, the group on the complex projective line starts with setting G=GL2⁡(C){\displaystyle G=\operatorname {GL} _{2}(\mathbb {C} )}, the group of invertible 2 × 2 complex matrices, S=SL2⁡(C){\displaystyle S=\operatorname {SL} _{2}(\mathbb {C} )}, the subgroup of determinant 1 matrices, and N{\displaystyle N} the normal subgroup of scalar matrices C×I={(a00a):a∈C×}{\displaystyle \mathbb {C} ^{\times }\!I=\left\{\left({\begin{smallmatrix}a&0\\0&a\end{smallmatrix}}\right):a\in \mathbb {C} ^{\times }\right\}}, we have S∩N={±I}{\displaystyle S\cap N=\{\pm I\}}, where I{\displaystyle I} is the identity matrix, and SN=GL2⁡(C){\displaystyle SN=\operatorname {GL} _{2}(\mathbb {C} )}. Then the second isomorphism theorem states that: PGL2⁡(C):=GL2⁡(C)/(C×I)≅SL2⁡(C)/{±I}=:PSL2⁡(C){\displaystyle \operatorname {PGL} _{2}(\mathbb {C} ):=\operatorname {GL} _{2}\left(\mathbb {C} )/(\mathbb {C} ^{\times }\!I\right)\cong \operatorname {SL} _{2}(\mathbb {C} )/\{\pm I\}=:\operatorname {PSL} _{2}(\mathbb {C} )}Theorem C (groups) Let G{\displaystyle G} be a group, and N{\displaystyle N} a normal subgroup of G{\displaystyle G}. Then If K{\displaystyle K} is a subgroup of G{\displaystyle G} such that N⊆K⊆G{\displaystyle N\subseteq K\subseteq G}, then G/N{\displaystyle G/N} has a subgroup isomorphic to K/N{\displaystyle K/N}. Every subgroup of G/N{\displaystyle G/N} is of the form K/N{\displaystyle K/N} for some subgroup K{\displaystyle K} of G{\displaystyle G} such that N⊆K⊆G{\displaystyle N\subseteq K\subseteq G}. If K{\displaystyle K} is a normal subgroup of G{\displaystyle G} such that N⊆K⊆G{\displaystyle N\subseteq K\subseteq G}, then G/N{\displaystyle G/N} has a normal subgroup isomorphic to K/N{\displaystyle K/N}. Every normal subgroup of G/N{\displaystyle G/N} is of the form K/N{\displaystyle K/N} for some normal subgroup K{\displaystyle K} of G{\displaystyle G} such that N⊆K⊆G{\displaystyle N\subseteq K\subseteq G}. If K{\displaystyle K} is a normal subgroup of G{\displaystyle G} such that N⊆K⊆G{\displaystyle N\subseteq K\subseteq G}, then the quotient group (G/N)/(K/N){\displaystyle (G/N)/(K/N)} is isomorphic to G/K{\displaystyle G/K}.The last statement is sometimes referred to as the third isomorphism theorem. The first four statements are often subsumed under Theorem D below, and referred to as the lattice theorem, correspondence theorem, or fourth isomorphism theorem. Theorem D (groups) Let G{\displaystyle G} be a group, and N{\displaystyle N} a normal subgroup of G{\displaystyle G}. The canonical projection homomorphism G→G/N{\displaystyle G\rightarrow G/N} defines a bijective correspondence between the set of subgroups of G{\displaystyle G} containing N{\displaystyle N} and the set of (all) subgroups of G/N{\displaystyle G/N}. Under this correspondence normal subgroups correspond to normal subgroups. This theorem is sometimes called the correspondence theorem, the lattice theorem, and the fourth isomorphism theorem. The Zassenhaus lemma (also known as the butterfly lemma) is sometimes called the fourth isomorphism theorem. 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 f:G→H{\displaystyle f:G\rightarrow 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 ι∘π{\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 ker⁡f{\displaystyle \ker f} and a monomorphism κ:ker⁡f→G{\displaystyle \kappa :\ker f\rightarrow G} (kernels are always monomorphisms), which complete the short exact sequence running from the lower left to.... Discover the David J Cona popular books. Find the top 100 most popular David J Cona books.