Subgroup

Normal subgroup

A normal subgroup, also called an invariant subgroup, is a subgroup 𝐻 ⊆𝐺 whose only conjugate subgroup is itself¹, #m/def/group i.e. for all 𝑔 ∈𝐺 and ℎ ∈𝐻

This is often denoted as 𝐻 ⊴𝐺.

Every group has two trivial normal subgroups, {𝑒} and 𝐺. A finite group with no non-trivial normal subgroup is called a Simple group.

Alternative definition

Normal subgroups are sometimes given the following equivalent definition using cosets:²

A subgroup 𝐻 of a group 𝐺 is called a normal subgroup of 𝐺 iff. 𝑎𝐻 =𝐻𝑎 for all 𝑎 ∈𝐺, i.e. the left and right Coset in every element the same.

Properties

1. Normal subgroups uniquely specify all congruence relations on the group, see Correspondence between normal subgroups and congruence relations.
2. As a consequence of the above property, a normal subgroup 𝑁 ⊴𝐺 may be used to form a Quotient group 𝐺/𝑁 Indeed this construction is only possible if a subgroup is normal.
3. The intersection of normal subgroups is a normal subgroup.

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