First isomorphism theorem - Revision history

InfernalAtom683 at 13:34, 10 April 2026

2026-04-10T13:34:32Z

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 Let <math>G</math> and <math>H</math> be [[Group|groups]] and <math>f\colon g\to H</math> a group homomorphism. Then,
-* The kernel of <math>f</math>, <math>\ker f\trianglelefteq G</math> is a normal subgroup of <math>G</math>.
+# The kernel of <math>f</math>, <math>\ker f\trianglelefteq G</math> is a normal subgroup of <math>G</math>.
-* The image of <math>f</math>, <math>\operatorname{im}f<G</math> is a subgroup of <math>H</math>.
+# The image of <math>f</math>, <math>\operatorname{im}f<G</math> is a subgroup of <math>H</math>.
-* <math>G/\ker f \cong \operatorname{im} f</math>.
+=== Proof ===

InfernalAtom683 at 12:56, 10 April 2026

2026-04-10T12:56:14Z

← Older revision		Revision as of 20:56, 10 April 2026
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	Let <math>G</math> and <math>H</math> be [[Group\|groups]] and <math>f\colon g\to H</math> a group homomorphism. Then,		Let <math>G</math> and <math>H</math> be [[Group\|groups]] and <math>f\colon g\to H</math> a group homomorphism. Then,

	* The kernel of <math>f</math> is a normal subgroup of <math>G</math>.		* The kernel of <math>f</math>, <math>\ker f\trianglelefteq G</math> is a normal subgroup of <math>G</math>.
	* The image of <math>f</math> is a subgroup of <math>H</math>.		* The image of <math>f</math>, <math>\operatorname{im}f<G</math> is a subgroup of <math>H</math>.

	* <math>G/\ker f \cong \operatorname{im} f</math>.		* <math>G/\ker f \cong \operatorname{im} f</math>.

InfernalAtom683: /* Group theory */

2026-04-04T11:19:09Z

Group theory

← Older revision		Revision as of 19:19, 4 April 2026
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	* The image of <math>f</math> is a subgroup of <math>H</math>.		* The image of <math>f</math> is a subgroup of <math>H</math>.

	* <math>G/\ker f \cong \operatorname{im} f</math> .		* <math>G/\ker f \cong \operatorname{im} f</math>.

InfernalAtom683: Created page with "The '''first isomorphism theorem''' is a fundamental result in abstract algebra that describes the relationship between a homomorphism, its kernel, and its image. The theorem appears uniformly across algebraic structures such as groups, rings, and modules, and serves as a prototype for many structural results in algebra. Specifically, given a homeo..."

2026-04-04T11:18:24Z

Created page with "The '''first isomorphism theorem''' is a fundamental result in abstract algebra that describes the relationship between a homomorphism, its kernel, and its image. The theorem appears uniformly across algebraic structures such as groups, rings, and modules, and serves as a prototype for many structural results in algebra. Specifically, given a homeo..."

New page

The '''first isomorphism theorem''' is a fundamental result in [[abstract algebra]] that describes the relationship between a [[homomorphism]], its [[Kernel (algebra)|kernel]], and its [[Image (mathematics)|image]]. The theorem appears uniformly across algebraic structures such as [[Group (mathematics)|groups]], [[Ring (mathematics)|rings]], and [[Module (mathematics)|modules]], and serves as a prototype for many structural results in algebra. Specifically, given a homeomorphism, the quotient of its domain by its kernel is isomorphic to its image.

== Group theory ==

=== Statement ===
Let <math>G</math> and <math>H</math> be [[Group|groups]] and <math>f\colon g\to H</math> a group homomorphism. Then,

* The kernel of <math>f</math> is a normal subgroup of <math>G</math>.
* The image of <math>f</math> is a subgroup of <math>H</math>.

* <math>G/\ker f \cong \operatorname{im} f</math> .