Group isomorphism theorems

The isomorphism theorems for groups are expressed as follows

First isomorphism theorem

Let $𝜑 : 𝐺 \to 𝐻$ be a Group homomorphism. Then the quotient by the kernel is isomorphic to the image: #m/thm/group

𝐺 k e r 𝜑 ≅ i m 𝜑 \leq 𝐻

Let $𝐴, 𝐵 ⊴ 𝐺$ . Then #m/thm/group

𝐴 𝐵 𝐵 ≅ 𝐴 𝐴 \cap 𝐵

Let $𝐴, 𝐵 ⊴ 𝐺$ be normal subgroups so that $𝐴 \leq 𝐵$ . Then $𝐵 / 𝐴 ⊴ 𝐺 / 𝐴$ and #m/thm/group

𝐺 / 𝐴 𝐵 / 𝐴 ≅ 𝐺 𝐵

#state/tidy | #lang/en | #SemBr