In detail, if G is a group, then a chief series of G is a finite collection of normal subgroups Ni ⊆ G, such that each quotient group Ni+1/Ni, for i = 1, 2,..., n − 1, is a minimal normal subgroup of G/Ni.
In other words, a chief series may be thought of as "full" in the sense that no normal subgroup of G may be added to it.
In particular, a finite chief factor is a direct product of isomorphic simple groups.
For example, the group of integers Z with addition as the operation does not have a chief series.
Supposing there exists a chief series Ni leads to an immediate contradiction: N1 is cyclic and thus is generated by some integer a, however the subgroup generated by 2a is a nontrivial normal subgroup properly contained in N1, contradicting the definition of a chief series.