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bibkey: nikolov2011powers authors: Nikolay Nikolov and Dan Segal year: 2011 title: Powers in finite groups doi: 10.4171/GGD/136 url: https://doi.org/10.4171/GGD/136 claim: In a finite group G, if N is normal, G is generated by N together with X and an ordered n-tuple y, and n is at least the generator rank of G, then left multiplication of each coordinate of y by a suitable element of N gives generators together with the unchanged set X. strata_touched:

  • D5/S3/FiniteGroups/GaschutzFixed
  • D5/S3/FiniteGroups/NikolovSegal/SectionClosure
  • D5/S3/FiniteGroups/NikolovSegal/OrderedPowerProducts
  • D5/S3/FiniteGroups/NikolovSegal/BoundedTupleLift
  • D5/S3/FiniteGroups/NikolovSegal/FiniteNormalInduction
  • D5/S3/FiniteGroups/NikolovSegal/GeneratorDecomposition
  • D5/S3/FiniteGroups/NikolovSegal/MinimalNormalSocle
  • D5/S3/FiniteGroups/NikolovSegal/MinimalNormalStructure
  • D5/S3/FiniteGroups/NikolovSegal/MinimalNormalConjugacy
  • D5/S3/FiniteGroups/NikolovSegal/SimpleSectionsOfProducts
  • D5/S3/FiniteGroups/NikolovSegal/LargeMinimalNormalStructure
  • D5/S3/FiniteGroups/NikolovSegal/CosetPowerBridge
  • D5/S3/FiniteGroups/NikolovSegal/InvariantBlockReduction license: No redistribution license asserted for the paper; citation only triage: anchor

Gaschutz lifting with a fixed set

Nikolay Nikolov and Dan Segal, Powers in finite groups, Groups, Geometry, and Dynamics 5 (2011), 501–507, DOI https://doi.org/10.4171/GGD/136. Lemma 1 on printed page 504 states the following extension of Gaschutz’s generator-lifting lemma. For a finite group G, a normal subgroup N, a subset X of G and elements y1 through yn, suppose G equals the product of N with the subgroup generated by X and the yi, with n at least d(G). There are elements a1 through an of N such that X and the ordered products ai yi generate G. Here d(G) is the minimum size of a generating set.

The fixed set is arbitrary. No commutativity, solvability or restriction to the Frattini subgroup is assumed. The multiplication order is ai yi. The tuple condition that an ordered n-tuple generates G is equivalent to d(G) being at most n, with repetitions and identity padding allowed. At n equal to zero, this condition forces G to be trivial.

The proof uses a counting invariant. For a subgroup H containing X, the number of corrections placing every coordinate in H is zero if N and H do not generate G, and otherwise equals the size of the intersection of N and H raised to n. Normality supplies a correction in each coset; translation identifies its fiber with the intersection of N and H. Finite inclusion-exclusion over proper subgroups containing X makes the number of generating corrections independent of the quotient-generating tuple. A generating tuple has the identity correction, so every such quotient-generating tuple has a generating correction.

The finite inclusion-exclusion identity, subgroup product membership, cardinality of products of finite types and minimum-generator-rank APIs are supplied by Mathlib. The group-theoretic lifting conclusion is the literature result cited above. It is a finite-group lemma; it does not alone establish strong completeness of finitely generated profinite groups.

The paper attributes the original lifting result to W. Gaschutz, Zu einem von B. H. und H. Neumann gestellten Problem, Mathematische Nachrichten 14 (1955), 249–252, and points to Fried and Jarden, Field Arithmetic, Lemma 15.30, for a proof that adapts to this version.

Verified locator

  • DOI: 10.4171/GGD/136

  • URL: https://doi.org/10.4171/GGD/136

  • Locator: printed page 504, Lemma 1, the fixed-set extension of Gaschutz lifting.

  • Locator: printed page 504, Lemma 2, the alternating-section supplement. For every finite group G and normal subgroup N, some subgroup L satisfies N join L = G and alpha(L) ≤ max(alpha(G/N), 4). Here alpha is the largest alternating section degree, and a section is a quotient of an arbitrary subgroup. The same Sylow-normalizer supplement transfers every alternating section of degree at least five to G/N.

  • Related source DOI: 10.4007/annals.2007.165.171

  • Related source URL: https://doi.org/10.4007/annals.2007.165.171

  • Locator: Nikolov and Segal, On finitely generated profinite groups, I: strong completeness and uniform bounds, printed pages 227–228, equation (43), the triangular substitution and the invariant-block reduction of Proposition 10.1 to Proposition 10.2.

Proposition 3 and its absorption input

On printed page 504, Proposition 3 gives sets X and Y for a finite d-generator group G with |X| and |Y| at most d, G generated by their union, X contained in the products of exactly m qth powers, and the subgroup generated by Y in the class C of groups with alpha at most k. The constants m and k are fixed before G. Products are ordered; identity padding and repetitions are allowed. The induction on page 504 leads into the use of Proposition 3 on page 505.

The conditional finite-group statement uses arbitrary natural q, m and k with k at least four, every finite G and every natural d with Group.rank(G) at most d, including d equal to zero. It produces two tuples indexed by Fin d whose ranges jointly generate G. Each coordinate of the first tuple is a product of the same exact m ordered qth powers, and the subgroup generated by the second tuple has alpha at most k. Their finite images give the paper’s cardinality bounds without a separate mathematical claim.

The remaining explicit premise, LargeMinimalNormalAbsorption(q,m,k), says that for every finite group G and every minimal nontrivial normal subgroup N outside C, left multiplication by any element of N preserves that same exact power-product set. Its quantification is uniform over G with m already fixed. This premise has no supplied proof. In the paper, the large-case step follows from Proposition 2 and the classification of minimal normal subgroups: such an N is a direct product of nonabelian simple groups whose orders exceed the relevant bound C(q).

For the induction, quotienting by N strictly decreases finite order and does not increase generator rank. Power products lift factor by factor. The same Lemma 2 supplement lifts the quotient Y tuple inside C. When N lies in C, Gaschutz fixes X and corrects Y; subgroup and extension closure of C preserve the bound. When N lies outside C, Gaschutz fixes Y and corrects X; the explicit absorption premise preserves the exact power-product length. These are distinct cases of the original induction.

This conditional result supplies neither the absorption premise nor uniform finite-group power width. Proposition 1, Proposition 2, Proposition 4, the restricted Burnside order bound and the unconditional profinite conclusions are not consequences of this induction alone. The source is Powers in finite groups, DOI 10.4171/GGD/136; no mathematical novelty or resolution of an open problem is claimed.

Elementary structure before Proposition 2

Proposition 3, Case 1, on printed page 504 uses the structure of a minimal nontrivial normal subgroup N outside the bounded alternating-section class. For fixed natural C and k with k at least four and 2C less than k factorial, the elementary structural conclusion holds for every finite ambient group G and every such N with alpha(N) greater than k. N is perfect and centerless. Its actual minimal nontrivial normal subgroups form a finite family of nonabelian simple factors, each of order greater than C. Their internal product is isomorphic to N, and N modulo its center is isomorphic to that same product. Factor finiteness follows from ambient finiteness.

The native formalization proves these facts without the classification of finite simple groups. Characteristic subgroups of N are trivial or full. Its socle is characteristic and contains a minimal nontrivial normal subgroup, hence fills N. Distinct factors commute. A subgroup normal in one factor is normalized by that factor and centralized by all others, so it is normal in N; minimality then proves simplicity. Centerlessness gives full supremum independence, which makes the canonical product map injective. The full socle makes it surjective.

Ambient conjugates of any nontrivial subgroup of N join to N. This proves transitivity of ambient conjugation on the actual factors and supplies isomorphisms between them. A simple section of a finite product occurs in one factor even if its section subgroup is arbitrary. Thus the alternating section of degree n = alpha(N), with n at least five, occurs in one factor. Its order n factorial divided by two bounds that factor’s order from below. The fixed factorial inequality gives order greater than C, and conjugacy transfers the bound to every factor. No decomposition or conjugacy hypothesis is added. The abelian elementary-abelian decomposition is not needed for this conclusion.

The remaining power input is stronger than ordinary power width within each simple factor. For each positive q, it requires constants m(q) and C(q) fixed before G: whenever a normal subgroup N of arbitrary finite G is a product of nonabelian simple groups of order greater than C(q), every prescribed ordered tuple h of length m(q) and every a in N admit a tuple b in N of the same length satisfying

[ \prod_{i=0}^{m(q)-1}(b_i h_i)^q = a\prod_{i=0}^{m(q)-1}h_i^q. ]

This uniform prescribed-coset surjectivity is unproved here. It would imply the explicit LargeMinimalNormalAbsorption premise of the inherited generator induction; the elementary structure alone does not. The 2011 Proposition 2, printed page 503, permits perfect normal subgroups with a nontrivial center as well. The centerless specialization suffices at this minimal-normal step.

The deep supplier is Nikolov and Segal, On finitely generated profinite groups, I: strong completeness and uniform bounds, Annals of Mathematics 165 (2007), 171–238, Proposition 10.1, printed pages 226–232, DOI https://doi.org/10.4007/annals.2007.165.171. Its proof uses Proposition 10.2 and the uniform twisted-commutator and automorphism-power results proved in Part II, Products in quasisimple groups, Annals of Mathematics 165 (2007), 239–273, Theorems 1.1 and 1.2, DOI https://doi.org/10.4007/annals.2007.165.239. These citations supply no imported Lean theorem. Uniform coset surjectivity, the remaining uniform width and acceptable-subgroup results, restricted Burnside bounds, and unconditional strong completeness remain outside this formalization.

Algebraic reduction of prescribed coset powers

The reduction on printed pages 227–228 of Part I works in an arbitrary group G with any normal subgroup N. Fix natural q and m and an arbitrary prescribed tuple h indexed by Fin m. Products retain increasing index order. Use the paper’s conventions x^g = g inverse times x times g and [x,y] = x inverse times y inverse times x times y. Let tau(i,h) be the inverse of the ordered product of h(j)^q for j less than i. The corrected element is (h(i) inverse)^tau(i,h), and the corresponding automorphism of N is Mathlib’s normal conjugation by its inverse.

For tuples x and b, set a(i) = x(i)^b(i) [b(i),h(i) inverse]. Equation (43) states that psi(a) is the ordered product of [b(i),((x(i)h(i))^q) inverse]^tau(i,xh), followed by psi(x) on the right, where psi(x) = product((x(i)h(i))^q) product(h(i)^q) inverse. The inverse prefixes and the rightmost psi(x) are essential in noncommutative groups. The corrected automorphisms generate exactly the action group generated by the original normal conjugations by h(i), so properties of that action group transfer to the corrected tuple.

The independent premise PrescribedCommutatorCoverage(A,q,m,k) requires an inner tuple y chosen before every target t in A, such that some tuple c has ordered product

[ \prod_{i=0}^{m-1} c_i^{-1} \bigl(k_i\circ\operatorname{conj}(y_i^{-1})\bigr)^q(c_i)=t. ]

Here conj is Mathlib’s left conjugation, and composition applies the rightmost automorphism first. This is coverage by actual ordered commutator values, not generation of a commutator subgroup. Recursive triangular transport realizes every chosen inner tuple in N by coset elements x(i)h(i): at each position, only earlier coordinates determine the prefix. Their images in G/N agree with the prescribed prefix, which puts the required correction in N.

Under that value-set premise for the corrected automorphisms, equation (43) gives, for every a in N, a tuple b in N satisfying

[ \prod_{i=0}^{m-1}(b_i h_i)^q = a\prod_{i=0}^{m-1}h_i^q. ]

The exponent q and exact length m are unchanged. No finiteness, simple factor decomposition, positivity of q or restriction on the prescribed tuple is added to this conditional algebraic theorem. With the earlier large-minimal-normal structural data, a uniform value-set premise would therefore supply the absorption used by the generator induction.

For an explicitly given equivariant isomorphism A with a product of groups S(r), value-set coverage in every invariant block with the same q and m combines to value-set coverage in A. The proof chooses the inner tuples before targets, combines them through the isomorphism, and transports automorphism powers and ordered products coordinatewise. The index type and factor groups need not be finite. Construction of the actual factor-orbit splitting and its equivariance is not supplied by this conditional product theorem.

These are the algebraic reductions in Part I, DOI 10.4007/annals.2007.165.171, printed pages 227–228. The uniform transitive Proposition 10.2 theorem, quantitative simple-group inputs, uniform power width, restricted Burnside bounds and unconditional strong completeness remain unproved. In particular no group-, rank-, factor- or tuple-dependent choice of m establishes the required uniform theorem.

Actual factor orbits and conditional transitive coverage

For a prescribed automorphism tuple k, automorphisms act by subgroup image on the actual minimal normal factors. The group generated by k partitions those factors into orbits. Restricting to each orbit gives a genuine dependent product block and an automorphism homomorphism. The permutations generated by the restricted tuple equal the exact permutation image of the generated group and act transitively on that orbit. For a finite centerless group whose minimal normal socle is the whole group, the internal product regrouped by these orbits is an equivariant isomorphism with the group itself. No global transitivity or assumed orbit splitting is needed.

The independent transitive supplier fixes natural q, m and C and asks for ordered commutator-value coverage on every finite nonempty power of every nonabelian finite simple group of order greater than C. A prescribed action must have genuine component automorphisms and permutations whose generated group is transitive. The inner correction tuple is chosen before all targets: its value-set condition has quantifier order exists y, for every t, exists c. The uniform supplier chooses one positive m and C for each positive q before the simple group, factor count, index type and prescribed tuple. It is UNPROVED in this formal development.

For a minimal nontrivial normal subgroup N of a finite ambient group, ambient factor conjugacy transports each actual orbit block to a homogeneous simple power with its actual component automorphisms. Assuming the transitive supplier and that all factors are nonabelian simple of order greater than C, block coverage combines to coverage on N with the same q and exact m. With cutoff at least four and twice C less than cutoff factorial, the inherited large-alpha structure provides those factor hypotheses. The prescribed coset-power bridge then proves LargeMinimalNormalAbsorption(q,m,cutoff), which is consumed by the existing generator decomposition. No target surjectivity premise is added to the conditional reduction.

This construction and reduction follow Nikolov and Segal, On finitely generated profinite groups, I: strong completeness and uniform bounds, Annals of Mathematics 165 (2007), printed pages 227–228, DOI https://doi.org/10.4007/annals.2007.165.171. Published Proposition 10.2 also treats quasisimple groups and every length m at least z(q). The chosen- length centerless simple-group supplier here is a sufficient specialization, not a formalization of that full printed proposition and not an open research conjecture. Uniform power width, restricted Burnside bounds and unconditional strong completeness remain unproved in this development.