What is a normal subgroup? What natural map does it determine? How does this relate to covering spaces?
Solution
A subgroup \(N\le G\) is normal, written \(N\normal G\), if \[ gNg^{-1}=N\qquad\text{for every }g\in G. \] Equivalently, the left and right cosets of \(N\) coincide.
Normality is exactly the condition that makes the quotient set \(G/N\) into a group. The natural quotient homomorphism is \[ \pi:G\twoheadrightarrow G/N, \qquad g\longmapsto gN, \] and \[ \ker\pi=N. \] Conversely, kernels of group homomorphisms are normal, so normal subgroups are precisely kernels.
There is also a covering-space interpretation. Let \(X\) be path connected, locally path connected, and semilocally simply connected, and identify \[ G=\pi_1(X,x_0). \] Subgroups \(H\le G\) correspond to connected covering spaces of \(X\) up to equivalence. The covering corresponding to \(H\) is regular (Galois) exactly when \(H\normal G\). In that case its deck transformation group is naturally \[ G/H. \] Thus the algebraic triple \[ H\normal G\twoheadrightarrow G/H \] models the fundamental group of the covering, the fundamental group of the base, and the deck group of a regular covering.