# Isomorphism theorems

Three theorems relating to homomorphisms of general algebraic systems.

## First Isomorphism Theorem

Let $f : A \rightarrow B$ be a homomorphism of $\Omega$-algebras and $Q$ the kernel of $f$, as an equivalence relation on $A$. Then $Q$ is a congruence on $A$, and $f$ can be factorised as $f = \epsilon f' \mu$ where $\epsilon : A \rightarrow A/Q$ is the quotient map, $f' : A/Q \rightarrow f(A)$ and $\mu : f(A) \rightarrow B$ is the inclusion map. The theorem asserts that $f'$ is well-defined and an isomorphism.

## Second Isomorphism Theorem

Let $Q$ be a congruence on the $\Omega$-algebra $A$ and let $A_1$ be a subalgebra of $A$. The saturation $A_1^Q$ is a subalgebra of $A$, the restriction $Q_1 = Q \ \cap A_1 \times A_1$ is a congruence on $A_1$ and there is an isomorphism $$A_1 / Q_1 \cong A_1^Q / Q \ .$$

## Third Isomorphism Theorem

Let $A$ be an $\Omega$-algebra and $Q \subset R$ congruences on $A$. There is a unique homomorphism $\theta$ from $A/Q \rightarrow A/R$ compatible with the quotient maps from $A$ to $A/R$ and $A/Q$. If $R/Q$ denotes the kernel of $\theta$ on $A/Q$ then there is an isomorphism $$(A/Q)/(R/Q) \cong A/R \ .$$

## Application to groups

In the case of groups, a congruence $Q$ on $G$ is determined by the congruence class $N = [1_G]_Q$ of the identity $1_G$, which is a normal subgroup, and the other $Q$-classes are the cosets of $N$. It is conventional to write $G/N$ for $G/Q$. The saturation of a subgroup $H$ is the complex $H^Q = HN$.

### First Isomorphism Theorem for groups

Let $f : A \rightarrow B$ be a homomorphism of groups and $N = f^{-1}(1_B)$ the kernel of $f$. Then $N$ is a normal subgroup of $A$, and $f$ can be factorised as $f = \epsilon f' \mu$ where $\epsilon : A \rightarrow A/N$ is the quotient map, $f' : A/N \rightarrow f(A)$ and $\mu : f(A) \rightarrow B$ is the inclusion map. The theorem asserts that $f'$ is well-defined and an isomorphism.

### Second Isomorphism Theorem for groups

Let $N$ be a normal subgroup $A$ and let $A_1$ be a subgroup of $A$. The complex $NA_1$ is a subgroup of $A$, the intersection $N_1 = N \cap A_1$ is a normal subgroup o $A_1$ and there is an isomorphism $$A_1 / N_1 \cong A_1N/N \ .$$

### Third Isomorphism Theorem for groups

Let $A$ be a group and $N \subset M$ normal subgroups of $A$. There is a unique homomorphism $\theta$ from $A/N \rightarrow A/M$ compatible with the quotient maps from $A$ to $A/N$ and $A/M$. The set $M/N$ is the kernel of $\theta$ and hence a normal subgroup of $A/N$ and there is an isomorphism $$(A/N)/(M/N) \cong A/M \ .$$

How to Cite This Entry:
Richard Pinch/sandbox-2. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Richard_Pinch/sandbox-2&oldid=39628