Isomorphism theorems

Group isomorphism theorems

The isomorphism theorems for groups are expressed as follows

First isomorphism theorem

Let 𝜑 :𝐺 →𝐻 be a Group homomorphism. Then the quotient by the kernel is isomorphic to the image: group

𝐺ker⁡𝜑≅im⁡𝜑≤𝐻

Second isomorphism theorem

Let 𝐴,𝐵 ⊴𝐺. Then group

𝐴𝐵𝐵≅𝐴𝐴∩𝐵

Third isomorphism theorem

Let 𝐴,𝐵 ⊴𝐺 be normal subgroups so that 𝐴 ≤𝐵. Then 𝐵/𝐴 ⊴𝐺/𝐴 and group

𝐺/𝐴𝐵/𝐴≅𝐺𝐵

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