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Home / Modern Abstract Algebra / A subgroup H of group G is a normal subgroup if and only if aH = Ha, ∀ a ∈ G

A subgroup H of group G is a normal subgroup if and only if aH = Ha, ∀ a ∈ G

i.e., Every left coset of H in G is a right coset of H in G.

Proof: Let H be a normal subgroup of G.

Then g-1hg ∈ H, ∀ g ∈ G and ∀ h ∈ H.

Let a ∈ G and h ∈ H. Then a-1 ha ∈ H.

Thus a-1 ha = h1 for some h1 ∈ H.

⟹ ha = ah1 ∈ aH

Thus ha ∈ Ha ⟹ ha ∈ aH, and so Ha ⊂ aH

Now let ah ⊂ aH with h ∈ H.

Then ah = ah a-1a = (aha-1)a = ((a-1)-1ha-1)a ∈ Ha,

since (a-1)-1ha-1∈ Ha

Thus aH ⊂ Ha.

Hence aH = Ha, ∀ a ∈ G.

Conversely let aH =Ha , ∀ a ∈ G.

Let g ∈ G and h ∈ H.

Then g-1hg = g-1(hg).

Now hg ∈ Hg = gH ⟹ hg = g h1 for some h1 ∈ H.

[∵ g-1hg = g-1(gh1) = (g-1g)h1 = eh1 = h1]

This implies that g-1hg ∈ H.

Hence H is a normal subgroup of G. (Proved)

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