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third isomorphism theorem (Theorem)

If $ G$ is a group (or ring, or module) and $ H$ and $ K$ are normal subgroups (or ideals, or submodules, respectively) of $ G$, with $ H\subseteq K$, then there is a natural isomorphism $ (G/H)/(K/H)\cong G/K$.

This is usually known either as the Third Isomorphism Theorem, or as the Second Isomorphism Theorem (depending on the order in which the theorems are introduced). It is also occasionally called the Freshman Theorem.



"third isomorphism theorem" is owned by yark. [ full author list (2) | owner history (1) ]
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Other names:  freshman theorem

Attachments:
proof of third isomorphism theorem (Proof) by Thomas Heye
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Cross-references: second isomorphism theorem, natural isomorphism, submodules, ideals, normal subgroups, module, ring, group
There are 2 references to this entry.

This is version 8 of third isomorphism theorem, born on 2001-12-21, modified 2006-04-21.
Object id is 1126, canonical name is ThirdIsomorphismTheorem.
Accessed 6257 times total.

Classification:
AMS MSC: 20A05 (Group theory and generalizations :: Foundations :: Axiomatics and elementary properties)
 13A15 (Commutative rings and algebras :: General commutative ring theory :: Ideals; multiplicative ideal theory)
 16D10 (Associative rings and algebras :: Modules, bimodules and ideals :: General module theory)
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