Subgroup

Normal subgroup

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

𝑔ℎ𝑔−1∈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.

Proof of equivalence of definitions

Clearly

𝑔𝐻𝑔−1=𝐻∀𝑔∈𝐺⟺𝑔𝐻=𝐻𝑔∀𝑔∈𝐺

Hence the two definitions are equivalent.

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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Footnotes

1. 2023, Groups and representations, p. 13 ↩

2. 2017, Contemporary abstract algebra, p. 174 ↩