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Isomorphism Theorems

 First Isomorphism Theorem

Let πœ‘: 𝐺 ⟶ 𝐺′ be an epimorphism from 𝐺 to 𝐺′. Then:

a) The 𝐾 = π‘˜π‘’π‘Ÿπœ‘ is a normal subgroup of 𝐺.

b) The factor group 𝐺𝐾is isomorphic to 𝐺′.

c) A subgroup 𝐻′ of 𝐺′is normal in 𝐺′

if and only if its inverse image 𝐻 = πœ‘−1(𝐻′) is normal in 𝐺.

d) There is one-one correspondence between the subgroups of 𝐺′

and those subgroups of 𝐺 which contain π‘˜π‘’π‘Ÿπœ‘

Second Isomorphism Theorem

Let 𝐻 be a subgroup and 𝐾 be a normal subgroup of a group 𝐺, then;

a) 𝐻𝐾 is a subgroup of 𝐺,

b) 𝐻 ∩ 𝐾 is normal in 𝐻, and

c) 𝐻𝐾/𝐾≅𝐻/𝐻∩𝐾

Third Isomorphism Theorem

Let 𝐻,𝐾 be normal subgroups of a group 𝐺 and 𝐻 ⊆ 𝐾. Then 

(𝐺 𝐻) /(𝐾 𝐻) ≅ 𝐺 /𝐾.

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