Composition Series, Solvable Groups and Nilpotent Groups
Composition Series, Solvable Groups and Nilpotent Groups: core definitions, structural results and verification methods in abstract algebra.
How the topic fits together
Composition Series
This source section supplies the definitions, constructions and formal results used in this article. The sequence of results is preserved so later claims are not detached from their prerequisites.
Solvable And Nilpotent Groups
This source section supplies the definitions, constructions and formal results used in this article. The sequence of results is preserved so later claims are not detached from their prerequisites.
Core definitions and structural results
The following cards retain the mathematical content of the source while condensing long proofs into verification strategies. Numerical examples are treated as examples, not universal requirements.
Proposition
Suppose that the order of the finite group G is p2q, where p and q are distinct primes. Then G has either a normal Sylow p-subgroup or a normal Sylow q-subgroup. Thus G is not simple.
Proof / verification strategy: Use subgroup generation, coset/orbit counting and divisibility of orders. For cyclic groups, reduce the claim to arithmetic on exponents modulo the group order.
Definition
One way to break down a group into simpler components is via a subnormal series 1 = G0 ⊴G1 ⊴· · · ⊴Gr = G. “Subnormal” means that each subgroup Gi is normal in its successor Gi+1. In a normal series, the Gi are required to be normal subgroups of the entire group G. For convenience, the trivial subgroup {1} will be written as 1. Suppose that Gi is not a maximal normal subgroup of Gi+1, equivalently (by the correspondence theorem) Gi+1/Gi is not simple.
Proof / verification strategy: Use this as a definition checklist: identify the ambient object, test each stated condition separately, then compare examples and non-examples without assuming later theorems.
Lemma
(i) If K ⊴H ≤G and f is a homomorphism on G, then f(K) ⊴f(H). (ii) If K ⊴H ≤G and N ⊴G, then NK ⊴NH. (iii) If A, B, C and D are subgroups of G with A ⊴B and C ⊴D, then A(B∩C) ⊴A(B∩D), and by symmetry, C(D ∩A) ⊴C(D ∩B). (iv) In (iii), A(B ∩C) ∩B ∩D = C(D ∩A) ∩D ∩B.
Proof / verification strategy: Work directly from closure, inverses, subgroup tests and the defining action or presentation. Use a small cyclic or permutation model to verify the direction of the argument.
Zassenhaus Lemma Let A, B, C and D be subgroups of G, with A ⊴B and C ⊴D.
Zassenhaus Lemma Let A, B, C and D be subgroups of G, with A ⊴B and C ⊴D. Then A(B ∩D) A(B ∩C) ∼= C(D ∩B) C(D ∩A).
Proof / verification strategy: Work directly from closure, inverses, subgroup tests and the defining action or presentation. Use a small cyclic or permutation model to verify the direction of the argument.
Definition
If a subnormal series is refined by inserting H between Gi and Gi+1, consider allow H to coincide with Gi or Gi+1. If all such insertions are strictly between the “endgroups”, we will speak of a proper refinement. Two series are equivalent if they have the same length and their factor groups are the same, up to isomorphism and rearrangement.
Proof / verification strategy: Use this as a definition checklist: identify the ambient object, test each stated condition separately, then compare examples and non-examples without assuming later theorems.
Schreier Refinement Theorem
Schreier Refinement Theorem Let 1 = H0 ⊴H1 ⊴· · · ⊴Hr = G and 1 = K0 ⊴K1 ⊴· · · ⊴Ks = G be two subnormal series for the group G. Then the series have equivalent refinements.
Proof / verification strategy: Work directly from closure, inverses, subgroup tests and the defining action or presentation. Use a small cyclic or permutation model to verify the direction of the argument.
Jordan-H¨older Theorem
Jordan-H¨older Theorem If G has a composition series S (in particular if G is finite), then any subnormal series R without repetition can be refined to a composition series. Furthermore, any two composition series for G are equivalent.
Proof / verification strategy: Use subgroup generation, coset/orbit counting and divisibility of orders. For cyclic groups, reduce the claim to arithmetic on exponents modulo the group order.
Definition
A subgroup H of the group G is characteristic (in G) if for each automorphism f of G, f(H) = H. Thus f restricted to H is an automorphism of H. Consequently, if H is characteristic in G, then it is normal in G. If follows from the definition that if H is characteristic in K and K is characteristic in G, then H is characteristic in G.
Proof / verification strategy: Use this as a definition checklist: identify the ambient object, test each stated condition separately, then compare examples and non-examples without assuming later theorems.
More Definitions and Comments The commutator subgroup G′ of a group G is
More Definitions and Comments The commutator subgroup G′ of a group G is the subgroup generated by all commutators [x, y] = xyx−1y−1. (Since [x, y]−1 = [y, x], G′ consists of all finite products of commutators.) Here are some basic properties. (2) G′ is characteristic in G. This follows because any automorphism f maps a commutator to a commutator: f[x, y] = [f(x), f(y)].
Proof / verification strategy: Use this as a definition checklist: identify the ambient object, test each stated condition separately, then compare examples and non-examples without assuming later theorems.
Proposition
The following conditions are equivalent. (i) G is solvable. (ii) G has a normal series with abelian factors. (iii) G has a subnormal series with abelian factors.
Proof / verification strategy: Work directly from closure, inverses, subgroup tests and the defining action or presentation. Use a small cyclic or permutation model to verify the direction of the argument.
Proposition Subgroups and quotients of a solvable group are solvable. Conversely,
Subgroups and quotients of a solvable group are solvable. Conversely, if N is normal subgroup of G and both N and G/N are solvable, then G is solvable.
Proof / verification strategy: Work directly from closure, inverses, subgroup tests and the defining action or presentation. Use a small cyclic or permutation model to verify the direction of the argument.
Corollary
If G has a composition series, in particular if G is finite, then G is solvable if and only if the composition factors of G are cyclic of prime order.
Proof / verification strategy: Use subgroup generation, coset/orbit counting and divisibility of orders. For cyclic groups, reduce the claim to arithmetic on exponents modulo the group order.
Proposition
If G is a finite group, the following conditions are equivalent, and define a nilpotent group. [Nilpotence of an arbitrary group will be defined in (5.7.8).] (a) G is the direct product of its Sylow subgroups. (b) Every Sylow subgroup of G is normal.
Proof / verification strategy: Use subgroup generation, coset/orbit counting and divisibility of orders. For cyclic groups, reduce the claim to arithmetic on exponents modulo the group order.
Corollary
Every finite abelian group and every finite p-group is nilpotent.
Proof / verification strategy: Use subgroup generation, coset/orbit counting and divisibility of orders. For cyclic groups, reduce the claim to arithmetic on exponents modulo the group order.
Definition
A central series for G is a normal series 1 = G0 ⊴ G1 ⊴· · · ⊴Gr = G such that Gi/Gi−1 ⊆Z(G/Gi−1) for every i = 1, . . . , r. (The series just discussed is a special case called the upper central series.) An arbitrary group G is called nilpotent if it has a central series. Thus a finite p-group is nilpotent, and in particular, every Sylow p-subgroup is nilpotent. Now a direct product of a finite number of nilpotent groups is nilpotent.
Proof / verification strategy: Use this as a definition checklist: identify the ambient object, test each stated condition separately, then compare examples and non-examples without assuming later theorems.
Quick-reference relationships
Problem-solving workflow
Identify the ambient group
State the operation, identity, inverses and whether commutativity is available.
Locate the relevant subgroup structure
Check generated subgroups, normality, cosets, stabilisers or direct factors before applying a theorem.
Use the correct counting or mapping tool
Choose coset counting, orbit counting, a homomorphism, a quotient or a group action as appropriate.
Check hypotheses explicitly
Finite-order assumptions, normality, prime-power divisibility and coprimality conditions are not interchangeable.
Translate the result back to structure
Interpret the result as a statement about subgroup size, quotient structure, action, decomposition or presentation.
Verify with a small model
Use a cyclic, symmetric, dihedral or modular example to test the logic without treating the example as the theorem.
Worked-solution emphasis from the supplied source
The supplied worked solutions for this section repeatedly test order, subgroup, prime, factor, norm, Tor. These checks are used here as verification themes rather than copied as answer text.
Common mistakes and boundary conditions
- Treating left and right cosets as identical without normality.
- Assuming the converse of a subgroup-order divisibility result.
- Confusing the order of a group with the order of one of its elements.
- Using quotient multiplication before checking that the subgroup is normal.
Verification checklist
- State the ambient algebraic structure and operation before applying a theorem.
- Record every hypothesis that controls the result: finiteness, commutativity, normality, primality, separability, exactness or other section-specific conditions.
- Distinguish a definition from a theorem that follows from it.
- Check whether a map is well-defined before using its kernel, image, inverse or induced map.
- Use a concrete example only as a check; do not promote an illustrative value or pattern to a universal rule.
- When a quotient, localisation or extension is constructed, identify the canonical map and what becomes equal, invertible or fixed.
Source coverage map
| Source section | Subject | PDF pages analysed |
|---|---|---|
| 5.6 | Composition Series | 96–98 |
| 5.7 | Solvable And Nilpotent Groups | 99–101 |
Related Mathematics pages
Source note: synthesised from the supplied abstract-algebra PDF. The complete 298-page file, including diagrams and worked solutions, was reviewed. Source-identifying author and bibliographic personal details are intentionally omitted. Formal proofs are condensed; the page does not claim requirements or values not supported by the supplied mathematics.