Isomorphism theorems

In mathematics, specifically abstract algebra, the isomorphism theorems (also known as Noether's isomorphism theorems) are theorems that describe the relationship among quotients, homomorphisms, and subobjects. Versions of the theorems exist for groups, rings, vector spaces, modules, Lie algebras, and 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. Three years later, B. L. van der Waerden published Moderne Algebra, an influential early abstract algebra textbook that helped standardize the structural treatment of groups, rings, and fields in which these theorems appear prominently.

Groups

We first present the isomorphism theorems of the groups.

Theorem A (groups)

Let

G

{\displaystyle G}

and

H

{\displaystyle H}

be groups, and let

f

:

G

→

H

{\displaystyle f:G\rightarrow H}

be a homomorphism. Then:

The kernel of

f

{\displaystyle f}

is a normal subgroup of

G

{\displaystyle G}

,

The image of

f

{\displaystyle f}

is a subgroup of

H

{\displaystyle H}

, and

The image of

f

{\displaystyle f}

is isomorphic to the quotient group

G

/

ker

⁡

f

{\displaystyle G/\ker f}

.

In particular, if

f

{\displaystyle f}

is surjective then

H

{\displaystyle H}

is isomorphic to

G

/

ker

⁡

f

{\displaystyle G/\ker f}

.

This theorem is usually called the first isomorphism theorem.