Noether's bound for polynomial invariants of finite groups

Domokos, M.; Hegedűs, Pál

doi:10.1007/s000130050426

Cited by 24 publications

(25 citation statements)

References 4 publications

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“…Indeed, some of the bounds given in the preceding example are valid in a more general context as the next result shows, see [14,51]. There is another set of improvements.…”

Section: Example 35 (Daniela Krausementioning

confidence: 74%

Degree bounds—An invitation to postmodern invariant theory

Neusel

2007

Topology and its Applications

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This is an invitation to invariant theory of finite groups; a field where methods and results from a wide range of mathematics merge to form a new exciting blend. We use the particular problem of finding degree bounds to illustrate this.Invariant theory is a relatively young field. Although we will give explicit definitions later, a brief discussion on its short but illustrious history can give insight into this area of mathematics:In 1773, J.L. Lagrange observed that the determinant of a binary quadratic form is an invariant polynomial under any linear transformation.Around 1800, Carl F. Gauss considered the general problem of the invariance of binary quadratic forms with integral coefficients under an action of the special linear group. However, it was George Boole who, in 1843, laid the foundations of invariant theory with his papers [5,6]. In the second half of the 19th century, invariant theory blossomed under the hands of people like Sir Arthur Cayley in England and James J. Sylvester in England and the US, Felix Klein in Germany, and many others. Indeed, the history of invariant theory in the 19th century is exciting and rich, and we recommend the survey articles by W. Franz Meyer [33,34], and of course the lovely lectures by Felix Klein [29].A culmination point of this early phase was Paul Gordan's proof that the invariant binary forms (under an action of the special linear group SL(2, C)) are finitely generated (see [22]). However, the early history of invariant theory ended abruptly with David Hilbert's proof of the finite generatedness of the SL(n, C)-invariants using, or better creating, completely new tools and paradigms (see [24]). Indeed, David Hilbert's viewpoint was earthshaking and rang in mathematical modernity. 2 However, the school around David Hilbert and later Emmy Noether was just too successful; they solved the current problems of invariant theory of finite groups, leaving behind a seemingly complete, and thus dead branch of mathematics (see [15] and the response [46]). M.D. Neusel / Topology and its Applications 154 (2007) 792-814 793On the other hand, modular invariant theory over finite fields was a field presenting problems and questions of a completely different type. These were studied by Leonard E. Dickson and his school at Chicago. Impressive as these results are (Leonard E. Dickson's collected works, 6 huge volumes [13], are a treasure box for everybody who works in the area) the tools available back then were soon exhausted, and so no wonder that also his algorithmic approach faded.However, modernity that had marginalized classical invariant theory at the same time brought new questions and problems to invariant theory. In particular, the entire field of modular invariant theory over finite fields was revitalized by algebraic topology. 3 Nowadays there are many other connections and overlaps with fields like group cohomology, 4 representation theory, algebraic combinatorics (see [49] for an idiosyncratic look back), commutative algebra, and number theory.Also, the electronic rev...

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“…Indeed, some of the bounds given in the preceding example are valid in a more general context as the next result shows, see [14,51]. There is another set of improvements.…”

Section: Example 35 (Daniela Krausementioning

confidence: 74%

Degree bounds—An invitation to postmodern invariant theory

Neusel

2007

Topology and its Applications

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“…Let G be a non-cyclic nilpotent group and p the smallest prime divisor of |G|, then β(G) ≤ |G| p + p − 1 except if p = 2 and G is a dicyclic group, in which case β(G) = 1 2 |G| + 2. Compared with the existing upper bound [6,7,11], our result is an improvement when G is a non-cyclic nilpotent group if p is large.…”

Section: Introductionmentioning

confidence: 75%

Erdős–Ginzburg–Ziv theorem and Noether number for C⋉C

Han

Zhang

2019

Journal of Number Theory

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Let G be a multiplicative finite group and S = a 1 · . . . · a k a sequence over G. We call S a product-one sequence if 1 = k i=1 a τ (i) holds for some permutation τ of {1, . . . , k}. The small Davenport constant d(G) is the maximal length of a product-one free sequence over G. For a subsethave received a lot of studies. The Noether number β(G) which is closely related to zero-sum theory is defined to be the maximal degree bound for the generators of the algebra of polynomial invariants. Let G ∼ = Cm ⋉ϕ Cmn, in this paper, we prove that E(G) = d(G) + |G| = m 2 n + m + mn − 2 and β(G) = d(G) + 1 = m + mn − 1. We also prove that s mnN (G) = m + 2mn − 2 and provide the upper bounds of η(G), s(G). Moreover, if G is a non-cyclic nilpotent group and p is the smallest prime divisor of |G|, we prove that β(G) ≤

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“…Schmid [14] proved that over the field of complex numbers β(G) = |G| holds only when G is cyclic. This was sharpened by Domokos and Hegedűs [4] (and later by Sezer [15] in positive coprime characteristic) to β(G) 3 4 |G| unless G is cyclic.…”

Section: Introductionmentioning

confidence: 94%

Finite groups with large Noether number are almost cyclic

Hegedűs¹,

Maróti²,

Pyber³

2019

Annales De l'Institut Fourier

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Noether, Fleischmann and Fogarty proved that if the characteristic of the underlying field does not divide the order |G| of a finite group G, then the polynomial invariants of G are generated by polynomials of degrees at most |G|. Let β(G) denote the largest indispensable degree in such generating sets. Cziszter and Domokos recently described finite groups G with |G|/β(G) at most 2. We prove an asymptotic extension of their result. Namely, |G|/β(G) is bounded for a finite group G if and only if G has a characteristic cyclic subgroup of bounded index. In the course of the proof we obtain the following surprising result. If S is a finite simple group of Lie type or a sporadic group then we have β(S) |S| 39/40 . We ask a number of questions motivated by our results.Résumé. Noether, Fleischmann et Fogarty ont montré que si le caractéristique du corps sous-jacent ne divise pas l'ordre |G| d'un groupe fini, alors l'anneau de polynomes invariants de G est engendré par des polynomes de degré au pluségalà |G|. Notons par β(G) le plus haut degré indispensable en un tel système de générateurs. Cziszter et Domokos ont récemment décrit les groupes finis G tels que |G|/β(G) est au pluségalà 2. Nous démontrons une extension asymptotique de leur résultat,à savoir que |G|/β(G) est borné pour un groupe fini G si et seulement s'il admet un sous-groupe caractéristique cyclique d'indice borné. Durant la démonstration nous trouvons le résultat surprenant suivant : si S est un groupe fini simple de type de Lie ou l'un des groupes sporadiques alors on a β(S) |S| 39/40 . Nous posonségalament quelques questions motivées par nos résultats.

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Noether's bound for polynomial invariants of finite groups

Cited by 24 publications

References 4 publications

Degree bounds—An invitation to postmodern invariant theory

Degree bounds—An invitation to postmodern invariant theory

Erdős–Ginzburg–Ziv theorem and Noether number for C⋉C

Finite groups with large Noether number are almost cyclic

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