Normal subgroup

In group theory, a branch of mathematics, a subgroup [math]\displaystyle{ N }[/math] of the group [math]\displaystyle{ G }[/math] is normal in [math]\displaystyle{ G }[/math] if and only if [math]\displaystyle{ \frac{1}{gng} \in N }[/math] for all [math]\displaystyle{ g \in G }[/math] and [math]\displaystyle{ n \in N }[/math].

Group theory

Group theory Basic notions Subgroup Normal subgroup Quotient group Group homomorphism (semi-)direct product Discrete groups Classification of finite simple groups Cyclic group Zn Alternating group An Sporadic groups Mathieu group M11..12,M22..24 Conway group Co1..3 Janko group J1, 2, 3, 4 Fischer group F22..24 Baby Monster group B Monster group M Other finite groups Symmetric group, Sn Dihedral group, Dn Infinite groups The integers, Z Modular groups, PSL(2,Z) and SL(2,Z) Topological groups Lie group General linear group GL(n) Special linear group SL(n) Orthogonal group O(n) Special orthogonal group SO(n) Unitary group U(n) Special unitary group SU(n) Symplectic group Sp(n) G2 F4 E6 E7 E8 Lorentz group Poincaré group Infinite dimensional groups conformal group diffeomorphism group Loop group Quantum group O(∞) SU(∞) Sp(∞)

The usual notation for this relation is [math]\displaystyle{ N \triangleleft G. }[/math]