Abstract algebra MOC

Isomorphism theorems

The isomorphism theorems are a set of four theorems, most generally statable in the language of universal algebra: the congruence relation and quotient. For particular examples, see

• Group isomorphism theorems
• Ring isomorphism theorems
• Module isomorphism theorems
• Lie algebra isomorphism theorems

First theorem

Let be an algebra homomorphism. Then is a subalgebra of , the relation is a congruence and and are isomorphic. algebra

Proof

proof

Second theorem

Let be an algebra, a subalgebra of , and be a congruence on . Let further be the restriction of to and

be the equivalence glasses under intersecting . Then algebra

1. is a congruence on
2. is a subalgebra of isomorphic to

Proof

proof

Third theorem

Let be an algebra and be congruences on such that . Then

is a congruence on and is isomorphic to . algebra

Proof

proof

Fourth isomorphism theorem

Also called the correspondence theorem

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