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Definition 7.4 Let G be a group, H a subgroup of G. If for all
g ∈ G
we have gH = Hg, then
 Let G be a
 group, H a subgroup of G. If for all g ∈ G we have gH = Hg, then we say H is a
 normal subgroup of G and write H Δ G.

Definition 7.17 let
H be a subgroup of a group G. Then
 let
 H be a subgroup of a group G. Then NG(H) = { g ∈ G gHg1
 = H} is called the normalizer of H in G.

Definition 8.4 Let H be a normal subgroup of G.
 Let H be a
 normal subgroup of G. Then the group consisting of the cosets of H in G under
 the operation (aH)(bH)=(ab)H is called the quotient group of G by H, written
 G/H.

Definition 8.12 Let :G→G’ be a homomorphism.
 Let :G→G’ be a homomorphism. the
 image (G)={(x) x ∈ G} is always a
 subgroup of G’, and will be equal to G’ itself iff is onto. In this case, G” is said to be a
 homomorphic image of G.

Definition 9.5 Given two groups G1 and G2
 Given two groups
 G1 and G2, the group G1xG2 with the
 operation defined in the preceding theorem is called the direct product of G1
 and G2.

