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First isomorphism theorem: Suppose θ: G → H is a group homomorphism. Then kerθ\triangleleft G and there is a group isomorphism [θ] : G/ kerθ→ Im (θ) so that θ factors as

G → G/ ker θ\overset - θ   → Im(θ) \hookrightarrow H.

In particular, note that G/ kerθ ≅ Im(θ) .
Second isomorphism theorem: Suppose H ≤ G and K \triangleleft G . Then HK ≤ G ; also K \triangleleft HK and H ∩G = ∅ . What is more,

H/ (H ∩K) ≅ HK/K.

Third isomorphism theorem: Suppose that N \triangleleft G . Then H → H/N is a bijection between subgroups H with N ≤ H ≤ G and the subgroups of G/N . Under this bijection, H \triangleleft G iff h/N \triangleleft G/N and so

G/N H/N ≅ G/H.