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 𝑓 :𝐴 →𝐵 be an algebra homomorphism. Then im⁡𝑓 is a subalgebra of 𝐴, the relation 𝑥 ≡𝑦 ⟺ 𝑓(𝑥) =𝑓(𝑦) is a congruence and im⁡𝑓 and 𝐴/ ≅ are isomorphic. algebra

Proof

proof

Second theorem

Let 𝐴 be an algebra, 𝐵 a subalgebra of 𝐴, and ≡ be a congruence on 𝐴. Let further ≡𝐵 = ≡ ∩(𝐵 ×𝐵) be the restriction of ≡ to 𝐵 and

[𝐵]≡={𝐾∈𝐴/≡:𝐾∩𝐵≠∅}

be the equivalence glasses under ≡ intersecting 𝐵. Then algebra

1. ≡𝐵 is a congruence on 𝐵
2. [𝐵]≡ is a subalgebra of 𝐴/ ≡ isomorphic to 𝐵/ ≡𝐵

Proof

proof

Third theorem

Let 𝐴 be an algebra and ≡, ≡′ be congruences on 𝐴 such that ≡′ ⊆ ≡. Then

≡/≡′={([𝑎]≡′,[𝑏]≡′):(𝑎,𝑏)∈≡}=[−]≡′∘≡∘[−]−1≡′

is a congruence on 𝐴/ ≡′ and 𝐴/ ≡ is isomorphic to (𝐴/ ≡′)/( ≡/ ≡′). algebra

Proof

proof

Fourth isomorphism theorem

Also called the correspondence theorem

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