Home > Isomorphism theorem

1 Groups

First we state the isomorphism theorems for groups, where they take a simpler form and state important properties of quotient groups (also called factor groups). All three involve " modding out" by a normal subgroup.

1.1 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.

1.2 Second isomorphism theorem

Let N be a normal subgroup of the group G, and let S be any subgroup. The intersection N ∩ S of N and S is a normal subgroup of S, N is a normal subgroup of the join NS of N and S, and S/(N ∩ S) is isomorphic to SN/N.

1.3 Third 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.

2 Rings and modules

The isomorphism theorems are also valid for modules over a fixed ringIn ring theory, a branch of abstract algebra, a ring is an algebraic structure in which addition and multiplication are defined and have similar properties to those familiar from the integers. History See Ring theory Definition and notation A ring is an a R (and therefore also for vector spaceThe fundamental concept in linear algebra is that of a vector space or linear space . This is a generalization of the set of all geometrical vectors and is used throughout modern mathematics. Formal definition A set V is a vector space over a field F (fors over a fixed fieldIn abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication, and division (except division by zero) may be performed and the associative, commutative, and distributive rules hold, which are famil). One has to replace the term "group" by "R-module", "subgroup" and "normal subgroup" by " submodule", and "factor group" by " factor module".

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

The notation for the join in both these cases is "S + N" instead of "SN".

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