Category:Isomorphism Theorems
This category contains pages concerning Isomorphism Theorems:
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:
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:
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:
Subcategories
This category has the following 4 subcategories, out of 4 total.
Pages in category "Isomorphism Theorems"
The following 20 pages are in this category, out of 20 total.