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Home / Modern Abstract Algebra / A subgroup H of a group G is a normal subgroup if and only if g-1 Hg = H, ∀ g ∈ G

A subgroup H of a group G is a normal subgroup if and only if g-1 Hg = H, ∀ g ∈ G

Proof: Let H be a normal subgroup og G.

Then g-1hg ∈ H, ∀ g ∈ G and ∀ h ∈ H.

Since g-1 Hg = {g-1 hg | g ∈ h and h ∈ H},

g-1 Hg ⊂ H   ————————(i)

Again for every h ∈ H and for every g ∈ G

We have h = g-1g hg-1g = g-1(ghg-1)g ∈ g –1 Hg, since gh g -1= (g -1)-1hg – 1∈ H

∴ H ⊂ g -1 Hg ———————–(ii)

From (i) and (ii) we get

g -1 Hg = H.

Conversely let g -1 Hg = H, ∀ g ∈ G

Then for every g ∈ G and for every h ∈ H,

we have g -1hg ∈ g -1Hg and so g-1hg ∈ H.

Hence H is normal in G. (Proved)

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