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Normal subgroup

Normal subgroups:

 Every group 𝐺 has at least two normal subgroups namely the identity {𝑒} and the group 𝐺 itself. The 

normal subgroups which are different from these two subgroups are called proper normal subgroups. 

All the subgroups of an abelian group are normal. The non-abelian groups all of whose subgroups are 

normal are called Hamiltonian Groups.

Examplese

a) The group 𝑄 = {±1, ±𝑖, ±𝑗, ±𝑘} of quaternions is such that it is non-abelian but every subgroup of 

𝑄 is normal.

b) The centre of any group is normal. Since

𝜁(𝐺) = {𝑎 ∈ 𝐺 ∶ 𝑎𝑔 = 𝑔𝑎, ∀ 𝑔 ∈ 𝐺}, therefore

𝑔𝜁(𝐺)𝑔−1 = {𝑎 ∈ 𝐺 ∶ 𝑔𝑎𝑔−1 = 𝑎, ∀ 𝑔 ∈ 𝐺}.

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