A chief series is a strict normal series of a group with no proper normal-series refinement. Its factors are minimal normal subgroups of the appropriate quotient groups. A chief factor of a finite soluble group is elementary abelian, whereas a composition factor of a finite soluble group is cyclic of prime order. A chief factor can therefore have order larger than a prime.
A hypothetical finite chief series in has a first nonzero term . The subgroup is normal in and strictly between zero and that term. This contradicts the defining maximality condition for the first chief factor. Finite groups always admit chief series, by choosing a maximal chain of their normal subgroups; this example shows why finiteness matters.
A chief factor of is a quotient with normal in and no -normal subgroup strictly between them. Equivalently, is a minimal normal subgroup of . Such factors are the successive factors of a chief series, and are direct products of isomorphic simple groups when is finite.
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