Isomorphism Theorems

Preface

There is no standard numbering for the Isomorphism Theorems. Different authors use different labellings.

Therefore, the following nomenclature is to a greater or lesser extent arbitrary, and not necessarily the most widely used or standard. Please take care.

First Isomorphism Theorem

Groups

Let $\phi: G_1 \to G_2$ be a group homomorphism.

Let $\map \ker \phi$ be the kernel of $\phi$.

Then:

$\Img \phi \cong G_1 / \map \ker \phi$

where $\cong$ denotes group isomorphism.

Rings

Let $\phi: R \to S$ be a ring homomorphism.

Let $\map \ker \phi$ be the kernel of $\phi$.

Then:

$\Img \phi \cong R / \map \ker \phi$

where $\cong$ denotes ring isomorphism.

Vector Spaces

Let $K$ be a field.

Let $X$ and $Y$ be vector spaces over $K$.

Let $T : X \to Y$ be a linear transformation.

Let $\ker T$ be the kernel of $T$.

Let $X/\ker T$ be the quotient vector space of $X$ modulo $\ker T$.

Then $X/\ker T$ is isomorphic to $\Img T$ as a vector space.

Topological Vector Spaces

Let $K$ be a topological field.

Let $\struct {X, \tau_X}$ and $\struct {Y, \tau_Y}$ be vector spaces over $K$.

Let $T : X \to Y$ be a continuous and open linear transformation.

Let $\ker T$ be the kernel of $T$.

Let $X/\ker T$ be the quotient topological vector space of $X$ modulo $\ker T$.

Then $X/\ker T$ is topologically isomorphic to $\Img T$.

Second Isomorphism Theorem

Groups

Let $G$ be a group, and let:

$(1): \quad H$ be a subgroup of $G$

$(2): \quad N$ be a normal subgroup of $G$.

Then:

$\dfrac H {H \cap N} \cong \dfrac {H N} N$

where $\cong$ denotes group isomorphism.

Rings

Let $R$ be a ring, and let:

$S$ be a subring of $R$

$J$ be an ideal of $R$.

Then:

$(1): \quad S + J$ is a subring of $R$

$(2): \quad J$ is an ideal of $S + J$

$(3): \quad S \cap J$ is an ideal of $S$

$(4): \quad \dfrac S {S \cap J} \cong \dfrac {S + J} J$

where $\cong$ denotes group isomorphism.

This result is also referred to by some sources as the first isomorphism theorem.

Third Isomorphism Theorem

Groups

Let $G$ be a group, and let:

$H, N$ be normal subgroups of $G$

$N$ be a subset of $H$.

Then:

$(1): \quad N$ is a normal subgroup of $H$

$(2): \quad H / N$ is a normal subgroup of $G / N$

where $H / N$ denotes the quotient group of $H$ by $N$

$(3): \quad \dfrac {G / N} {H / N} \cong \dfrac G H$

where $\cong$ denotes group isomorphism.

Rings

Let $R$ be a ring.

Let:

$J, K$ be ideals of $R$

$J$ be a subset of $K$.

Then:

$(1): \quad K / J$ is an ideal of $R / J$

where $K / J$ denotes the quotient ring of $K$ by $J$

$(2): \quad \dfrac {R / J} {K / J} \cong \dfrac R K$

where $\cong$ denotes ring isomorphism.

Fourth Isomorphism Theorem

Let $\phi: R \to S$ be a ring homomorphism.

Let $K = \map \ker \phi$ be the kernel of $\phi$.

Let $\mathbb K$ be the set of all subrings of $R$ which contain $K$ as a subset.

Let $\mathbb S$ be the set of all subrings of $\Img \phi$.

Let $\phi^\to: \powerset R \to \powerset S$ be the direct image mapping of $\phi$.

Then its restriction $\phi^\to: \mathbb K \to \mathbb S$ is a bijection.

Also:

$(1): \quad \phi^\to$ and its inverse both preserve subsets.

$(2): \quad \phi^\to$ and its inverse both preserve ideals:

If $J$ is an ideal of $R$, then $\map {\phi^\to} J$ is an ideal of $S$

If $J'$ is an ideal of $S$, then $\map {\paren {\phi^\to}^{-1} } {J'}$ is an ideal of $R$