Word fibres over finite chain rings
and the Amit–Ashurst conjecture for metacyclic groups
Henry Zweiman
September 9, 2026
Abstract
We prove the Amit–Ashurst conjecture for every finite metacyclic -group, including nonsplit extensions. The proof applies more generally to cyclic extensions of finite modules over commutative chain rings, when conjugation acts as multiplication by a principal unit. In particular, it also proves the conjecture for every finite -group with an elementary abelian normal subgroup and cyclic quotient. The main estimate concerns split extensions by abelian -groups. After fixing the quotient coordinates of a word, its base coordinate is linear in the module variables. Finite differences control the support of a coefficient of least valuation, and the module filtration compensates for the resulting loss of support. A stronger word-dependent estimate then descends through a central cyclic kernel, removing the splitting hypothesis for cyclic quotients. All bounds hold for every word and every attained value, without a bound on the number of variables or the nilpotency class.
1. Introduction and principal results
For a finite group , a word induces a map . Write for its image and
The Amit–Ashurst conjecture asserts that, for every finite nilpotent group ,
The formulation appears in Ashurst’s thesis [2, Conjecture 6.2.1] and is discussed under this name by Camina, Cocke and Thillaisundaram [3]. It strengthens the Amit conjecture, which concerns only the identity fibre. We always permit arbitrary , including words with zero exponent sums. The identity word causes no exception to (1).
A group is metacyclic if it has a cyclic normal subgroup with cyclic quotient. Our first main conclusion is the following.
Theorem 1.1.
Every finite nilpotent metacyclic group satisfies the Amit–Ashurst conjecture. In particular, for every finite metacyclic -group , every , and every ,
No splitting hypothesis is required.
Camina, Cocke and Thillaisundaram prove (1) for finite -groups with a cyclic maximal subgroup [3, Theorem A]. Their Sections 2–3 already use linearization after fixing cyclic coordinates and polynomial arguments. Theorem 1.1 removes the cyclic-maximal-subgroup restriction. For example, has class four and no cyclic maximal subgroup; a direct verification is given in Section 5. Levy’s split-extension result [5, Proposition 1.3] supplies the identity-fibre bound for split metacyclic groups. Controlling every attained value requires an additional argument.
The second main conclusion concerns modules of arbitrary rank.
Theorem 1.2.
Let be a finite -group with an elementary abelian normal subgroup such that is cyclic. Then satisfies the Amit–Ashurst conjecture.
In particular, this applies to for all and . Recent work of de las Heras, Toti and Vannacci [4] gives finer fibre estimates for selected Engel words in cyclic wreath products. Theorem 1.2 treats every word; it does not provide their finer estimates or settle their separate question about profinite sections.
Both results follow from a theorem about modules over finite chain rings. A finite commutative chain ring is a finite commutative ring with identity whose ideals form a chain. We write its ideals as
The field case has and . Modules are unital and finite. A principal unit is an element of .
Theorem 1.3.
Suppose a finite group contains the additive group of a finite -module as a normal subgroup, where is a finite commutative chain ring of residue characteristic . Suppose is cyclic of -power order and a lift of its generator acts on by multiplication by a principal unit of . Then satisfies (1).
The proof first establishes a sharper estimate for , where is any finite abelian -group acting by principal units. It then constructs a split cover of a cyclic extension. The extra strength in the split estimate is essential: no general assertion that (1) descends to arbitrary quotients is used.
The finite-difference language is the established functional-degree framework; see Aichinger and Moosbauer [1]. We give the elementary support estimate needed here in full. The contribution is its use with a common coefficient ideal on each quotient fibre, followed by the cyclic-cover transfer. The argument does not require polynomial coordinates on a finite abelian group or a choice of digits in a chain ring.
2. Finite differences and word coefficients
For finite abelian groups , a function , and , set
Say that has degree at most if every composition of difference operators annihilates it. This condition is meaningful without a field structure on either group.
Lemma 2.1.
If is nonzero and has degree at most , then
Proof.
For the function is a nonzero constant. More generally a nonzero constant satisfies the assertion for every . Otherwise choose such that . Since difference operators commute, has degree at most . Induction and the containment
give .
Now let be a finite commutative chain ring, let be a finite -module, and let be a finite abelian -group. Fix a homomorphism
Use additive notation for , and form with multiplication
For a word , let be the exponent sum of , and put
Lemma 2.2.
There are functions such that
Each is a finite signed sum of homomorphisms from the additive group to the multiplicative group .
Proof.
Write , with , and set
The inverse of is . Reading the product in (2) gives
Both and are additive homomorphisms. Formula (5) also covers the empty word, with all sums zero.
For a fixed , the first coordinate of (4) is a homomorphism . Uniform inputs therefore give a uniform distribution on its image. The difficulty is to control how often a sufficiently large common image occurs as varies in a prescribed fibre of .
3. The chain-ring estimate
The ideal valuation on is defined by and
In the following refined statement we assume is faithful. This entails no restriction on the ensuing bound (8): for a nonzero module one may replace by , again a chain ring with the same residue field. The action and group are unchanged.
Theorem 3.1.
With the preceding notation, let be faithful. For define
Then the word image above is exactly . If , then for every ,
If , the only value above is , with probability . Consequently, for arbitrary finite , every attained satisfies
Proof.
Fix , choose , and write . The fibre of is , and its probability in is . Put . If , all coefficients vanish on this fibre, which proves the asserted exceptional case.
Suppose . Choose an index for which the minimum in (6) is attained. The function
is nonzero and has degree at most . To see this, a summand of (5) restricted to the fibre has the form , with additive. For ,
Thus
The same assertion holds for the signed sum defining . Lemma 2.1 shows that at least tuples in have nonzero modulo . By minimality of , this means .
For every , all the coefficients in (4) lie in . Its base image is therefore contained in . At each of the tuples just counted, for a unit , so . The base map is then onto . In particular, the entire image above is exactly .
At each such tuple the conditional probability of any prescribed is . Multiplying by the fraction of these tuples and by the top-fibre probability gives
It remains to compare the module sizes. For , is a nonzero vector space over . Indeed, equality would iterate to , contradicting , which follows from faithfulness. Consequently,
This proves (7) and (8) for nonzero , including after the faithful reduction. For , the group is , the word map is , and its attained values all have probability .
Remark 3.2.
The coefficient itself, with values in , is the function to which the difference lemma is applied. No assertion is needed about the functional degree of a separately chosen -adic digit. Also, may have mixed cyclic exponents: the proof works on its actual additive group.
The lower bound is sharp uniformly over words, since is uniformly distributed. Estimate (7) records additional information when the coefficient ideal is deep. More importantly for nonsplit groups, (8) retains the image size instead of replacing it by .
4. Cyclic extensions and attained lifts
We prove Theorem 1.3. The case is abelian, so suppose and replace by . Let have order , and let lift a generator. Write for its scalar action. Since and is abelian, on , hence in by faithfulness.
Regard
as an element of the additive module, and write . We have . The order of is : an exponent killing must be divisible by , and precisely when divides . Form
The map
is a well-defined surjective homomorphism, where is identified with its normal subgroup in the displayed product. Its kernel is
This subgroup is central: and show that commutes with and with the cyclic complement. Thus is central cyclic of order , and .
Fix a word with exponent sums . Put
For , expressions of the form below mean . The image size of the exponent-sum map on the complement of is
For central elements and arbitrary , direct collection gives
Let , with if all , and set . The products on the right of (13) range over . Hence, if is attained, every element of is attained and projects to . Since is cyclic,
Now take . Lifting any input tuple attaining gives an attained . Uniform inputs in project to uniform inputs in , so
For finite the exponent difference is ; for it is at most . It is zero when . Thus (15) is at least , proving Theorem 1.3.
Remark 4.1.
The proof gives the stronger bound displayed in (15). In particular, if every exponent sum is divisible by , every attained value has probability at least . The order here is the order of the chosen lift . The ordinary bound would lose a factor and would not suffice for the argument.
First let be a finite metacyclic -group, with normal cyclic subgroup and cyclic quotient. If , the group is cyclic. Otherwise take and identify with its additive module. Conjugation by a lift of a quotient generator is multiplication by a unit . This automorphism has -power order. Its reduction in is therefore trivial, since that group has order . Hence , including for , and Theorem 1.3 applies.
For a finite nilpotent metacyclic group , choose a cyclic normal subgroup with cyclic quotient. Every Sylow subgroup is metacyclic: is cyclic and normal in , and embeds in . Finally, is the direct product of its Sylow subgroups, and word probabilities multiply across direct products. Multiplying the bounds for the Sylow components proves (1).
Suppose and let be the linear automorphism of the -space induced by a lift. Since is abelian, . Thus satisfies . Give its module structure over
The lift acts by the principal unit . Apply Theorem 1.3. This also covers ; if the assertion is immediate.
Corollary 5.1.
For every and , the regular wreath product satisfies (1) for every word. If its exponent sums are all divisible by , every attained value has probability at least the reciprocal of the order of the base group.
Proof.
The base is elementary abelian and the complement cyclic, so Theorem 1.2 proves the first assertion. For the second, use the split realization and (8), with .
The absence of a nilpotency-class bound is substantive. In , with , the regular base module is
where the complement generator acts as . Its commutator with has image , and successive lower-central terms are for . Thus the class is .
For a small example outside the cyclic-maximal-subgroup case, consider
This is defined since . Its lower-central terms have orders
because for . Every element has order dividing . Indeed, for ,
For , the sums modulo are respectively and ; for , they consist of four periods with sum . All are divisible by . A cyclic maximal subgroup would have order , impossible by the exponent bound. Thus this class-four example is covered by Theorem 1.1 but not by the cyclic-maximal-subgroup hypothesis.
The split theorem also permits mixed-length modules over and abelian complements of arbitrary rank, provided their actions are scalar principal units. Freeness of the module is not required. All these conclusions belong to the same chain-ring argument.
6. Scope and further questions
The proof singles out two properties. First, after fixing the quotient coordinates, a word is linear in the base variables. Second, all coefficient ideals are comparable, so a coefficient of least valuation supplies an image containing every other conditional image. Finite differences then ensure that this largest image occurs sufficiently often.
These properties suggest the next question: does every finite -group with an abelian normal subgroup and cyclic quotient satisfy (1)? Theorem 1.3 answers it when the action is represented by a principal unit over a chain ring. An arbitrary action on an abelian -group can instead generate a commutative ring with incomparable ideals. The common-image argument then does not follow from the present proof. Nor do we treat arbitrary nonsplit extensions by noncyclic abelian quotients. The general Amit–Ashurst conjecture remains unresolved here.
Computational checks.
The proofs above do not depend on computation. Accompanying scripts compare complete word distributions from literal multiplication with coefficient calculations and with pushforward through the split cover. The recorded checks comprise 22 models, 745 word distributions, 110,451,815 literal word evaluations, and 20,302 attained fibres. They include residue characteristics two and three, mixed module lengths, abelian complements of rank two, and nonsplit quaternion and module extensions. These finite checks test the implementations and small instances; they are not a substitute for the universal arguments.
[2] C. Ashurst, Fibres of words in finite groups, a probabilistic approach, Ph.D. thesis, University of Bath, 2012. University of Bath repository.
[3] R. D. Camina, W. L. Cocke and A. Thillaisundaram, The Amit–Ashurst conjecture for finite metacyclic -groups, Eur. J. Math. 9 (2023), article 46. doi:10.1007/s40879-023-00644-x.
[4] I. de las Heras, T. Toti and M. Vannacci, Engel probability in wreath products of -groups, preprint, July 2026. arXiv:2607.11605v1.
[5] M. Levy, On the probability of satisfying a word in nilpotent groups of class 2, preprint, 2011. arXiv:1101.4286v1.