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Journal articles on the topic 'Degree of nilpotence'

Author: Grafiati

Published: 10 March 2023

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1

Kireeva, Elena, and Vladimir Shchigolev. "The nilpotence degree of quantum Lie nilpotent algebras." International Journal of Algebra and Computation 28, no. 06 (2018): 1119–28. http://dx.doi.org/10.1142/s0218196718500492.

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We consider the quantum analog of the Lie commutator [Formula: see text] for an invertible element [Formula: see text] of the ground field and prove lower and upper bounds for the nilpotence degree of an associative algebra satisfying an identity of the form [Formula: see text].

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2

POWELL, GEOFFREY M. L. "The tensor product theorem for &∇tilde;-nilpotence and the dimension of unstable modules." Mathematical Proceedings of the Cambridge Philosophical Society 130, no. 3 (2001): 427–39. http://dx.doi.org/10.1017/s030500410100500x.

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Let [Fscr ] be the category of functors from the category of finite-dimensional [ ]2-vector spaces to [ ]2-vector spaces. The concept of &∇tilde;-nilpotence in the category [Fscr ] is used to define a ‘dimension’ for the category of analytic functors which has good properties. In particular, the paper shows that the tensor product F [otimes ] G of analytic functors which are respectively &∇tilde;s and &∇tilde;t nilpotent is &∇tilde;s+t − 1-nilpotent.The notion of &∇tilde;-nilpotence is extended to define a dimension in the category of unstable modules over the mod 2 Steenrod algebra, which is shown to coincide with the transcendence degree of an unstable Noetherian algebra over the Steenrod algebra.

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3

Croome, Sarah, та Mark L. Lewis. "𝑝-groups with exactly four codegrees". Journal of Group Theory 23, № 6 (2020): 1111–22. http://dx.doi.org/10.1515/jgth-2019-0073.

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AbstractLet G be a p-group, and let χ be an irreducible character of G. The codegree of χ is given by {\lvert G:\operatorname{ker}(\chi)\rvert/\chi(1)}. Du and Lewis have shown that a p-group with exactly three codegrees has nilpotence class at most 2. Here we investigate p-groups with exactly four codegrees. If, in addition to having exactly four codegrees, G has two irreducible character degrees, G has largest irreducible character degree {p^{2}}, {\lvert G:G^{\prime}\rvert=p^{2}}, or G has coclass at most 3, then G has nilpotence class at most 4. In the case of coclass at most 3, the order of G is bounded by {p^{7}}. With an additional hypothesis, we can extend this result to p-groups with four codegrees and coclass at most 6. In this case, the order of G is bounded by {p^{10}}.

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4

Minh, Pham Anh. "Any nilpotence degree occurs in mod-p cohomology rings of p-groups." Mathematische Zeitschrift 249, no. 2 (2004): 387–400. http://dx.doi.org/10.1007/s00209-004-0703-7.

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5

ŁOPUSZAŃSKI, JAN. "A REMARK ON B.R.S. TRANSFORMATIONS." International Journal of Modern Physics A 03, no. 11 (1988): 2589–600. http://dx.doi.org/10.1142/s0217751x88001077.

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We scrutinize transformations constructed by taking the most general expressions consisting of terms of forms of the same degree in the fields A10, ν01, [Formula: see text], Q12 and F20 as well as operations d10 and s01, where the subscripts denote the degree of the forms in the x—and in the group parameter spaces. Imposing the requirement of nilpotence for d and s as well as Faddeev-Popov charge conservation we get a unique form of the B.R.S. transformation.

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6

Petrov, E. P. "On the degree of minimal identity of a finitely generated algebra with a fixed nilpotence index." Sibirskie Elektronnye Matematicheskie Izvestiya 16 (August 6, 2019): 1028–35. http://dx.doi.org/10.33048/semi.2019.16.071.

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7

Filho, Augusto Reynol. "Nilpotent Spaces: Some Inequalities on Nilpotency Degrees." Proceedings of the American Mathematical Society 115, no. 2 (1992): 501. http://dx.doi.org/10.2307/2159274.

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8

Reynol Filho, Augusto. "Nilpotent spaces: some inequalities on nilpotency degrees." Proceedings of the American Mathematical Society 115, no. 2 (1992): 501. http://dx.doi.org/10.1090/s0002-9939-1992-1093597-8.

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9

PAJOOHESH, H., P. RODRIGUEZ та C. WADDELL. "NILPOTENT INNER DERIVATIONS ON SOME SUBRINGS OF Mn(ℝ)". Journal of Algebra and Its Applications 12, № 08 (2013): 1350045. http://dx.doi.org/10.1142/s021949881350045x.

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It is known that the degree of nilpotency of a nilpotent derivation on a prime ring including the ring of n × n matrices must be an odd number. In this article we introduce subrings of the ring of of n × n matrices that admit derivations with an even degree of nilpotency.

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10

POURMAHMOOD-AGHABABA, HASAN. "APPROXIMATELY BIPROJECTIVE BANACH ALGEBRAS AND NILPOTENT IDEALS." Bulletin of the Australian Mathematical Society 87, no. 1 (2012): 158–73. http://dx.doi.org/10.1017/s0004972712000251.

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AbstractBy introducing a new notion of approximate biprojectivity we show that nilpotent ideals in approximately amenable or pseudo-amenable Banach algebras, and nilpotent ideals with the nilpotency degree larger than two in biflat Banach algebras cannot have the special property which we call ‘property (𝔹)’ (Definition 5.2 below) and hence, as a consequence, they cannot be boundedly approximately complemented in those Banach algebras.

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11

Kudryavtseva, Ganna, and Volodymyr Mazorchuk. "On the Semigroup of Square Matrices." Algebra Colloquium 15, no. 01 (2008): 33–52. http://dx.doi.org/10.1142/s1005386708000047.

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We study the structure of nilpotent subsemigroups in the semigroup M(n,𝔽) of all n×n matrices over a field 𝔽 with respect to the operation of the usual matrix multiplication. We describe the maximal subsemigroups among the nilpotent subsemigroups of a fixed nilpotency degree and classify them up to isomorphism. We also describe isolated and completely isolated subsemigroups and conjugated elements in M(n,𝔽).

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12

Tyurin, D. N. "Generalization of the Artin-Hasse logarithm for the Milnor -groups of -rings." Sbornik: Mathematics 212, no. 12 (2021): 1746–64. http://dx.doi.org/10.1070/sm9520.

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Abstract Let be a -adically complete ring equipped with a -structure. We construct a functorial group homomorphism from the Milnor -group to the quotient of the -adic completion of the module of differential forms . This homomorphism is a -adic analogue of the Bloch map defined for the relative Milnor -groups of nilpotent extensions of rings of nilpotency degree for which the number is invertible. Bibliography: 12 titles.

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13

Tyurin, D. N. "Generalization of the Artin-Hasse logarithm for the Milnor -groups of -rings." Sbornik: Mathematics 212, no. 12 (2021): 1746–64. http://dx.doi.org/10.1070/sm9520.

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Abstract Let be a -adically complete ring equipped with a -structure. We construct a functorial group homomorphism from the Milnor -group to the quotient of the -adic completion of the module of differential forms . This homomorphism is a -adic analogue of the Bloch map defined for the relative Milnor -groups of nilpotent extensions of rings of nilpotency degree for which the number is invertible. Bibliography: 12 titles.

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14

GILMAN, ROBERT H., DEREK F. HOLT, and SARAH REES. "COMBING NILPOTENT AND POLYCYCLIC GROUPS." International Journal of Algebra and Computation 09, no. 02 (1999): 135–55. http://dx.doi.org/10.1142/s0218196799000102.

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The notable exclusions from the family of automatic groups are those nilpotent groups which are not virtually abelian, and the fundamental groups of compact 3-manifolds based on the Nil or Sol geometries. Of these, the 3-manifold groups have been shown by Bridson and Gilman to lie in a family of groups defined by conditions slightly more general than those of the automatic groups, i.e. to have combings which lie in the formal language class of indexed languages. In fact, the combings constructed by Bridson and Gilman for these groups can also be seen to be real-time languages (i.e. recognized by real-time Turing machines). This article investigates the situation for nilpotent and polycyclic groups. It is shown that a finitely generated class 2 nilpotent group with cyclic commutator subgroup is real-time combable, as are all 2 or 3-generated class 2 nilpotent groups, and groups in specific families of nilpotent groups (the finitely generated Heisenberg groups, groups of unipotent matrices over Z and the free class 2 nilpotent groups). Further, it is shown that any polycyclic-by-finite group embeds in a real-time combable group. All the combings constructed in the article are boundedly asynchronous, and those for nilpotent-by-finite groups have polynomially bounded length functions, of a degree equal to the nilpotency class, c; this verifies a polynomial upper bound on the Dehn functions of those groups of degree c+1.

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15

JUHÁSZ, TIBOR, and ENIKŐ TÓTH. "COMMUTATOR IDENTITIES ON SYMMETRIC ELEMENTS OF GROUP ALGEBRAS." Journal of Algebra and Its Applications 12, no. 08 (2013): 1350044. http://dx.doi.org/10.1142/s0219498813500448.

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Let K be a field of odd characteristic p, and let G be the direct product of a finite p-group P ≠ 1 and a Hamiltonian 2-group. We show that the set of symmetric elements (KG)* of the group algebra KG with respect to the involution of KG which inverts all elements of G, satisfies all Lie commutator identities of degree t(P) or more, where t(P) denotes the nilpotency index of the augmentation ideal of the group algebra KP. In addition, if P is powerful, then (KG)* satisfies no Lie commutator identity of degree less than t(P). Applying this result we get that (KG)* is Lie nilpotent and Lie solvable, and its Lie nilpotency index and Lie derived length are not greater than t(P) and ⌈ log 2 t(P)⌉, respectively, and these bounds are attained whenever P is a powerful group. The corresponding result on the set of symmetric units of KG is also obtained.

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16

Moghaddam, Mohammad Reza R., Ali Reza Salemkar, and Kazem Chiti. "n-Isoclinism Classes and n-Nilpotency Degree of Finite Groups." Algebra Colloquium 12, no. 02 (2005): 255–61. http://dx.doi.org/10.1142/s1005386705000246.

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Gallagher (1970) and Gustafson (1973) introduced the commutativity degree of a finite group. In this paper, we define the n-nilpotency degree of finite groups for n ≥ 1, and prove some results as Lescot (1995) does for a certain class of groups. In particular, it is shown that the n-isoclinism of finite groups preserves their n-nilpotency degrees. Finally, some sharper and more general upper bound than previously known is constructed for the commutativity degree of non-abelian finite groups.

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17

Slattery, Michael C. "Character degrees and nilpotence class in p-groups." Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics 57, no. 1 (1994): 76–80. http://dx.doi.org/10.1017/s1446788700036065.

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AbstractWork of Isaacs and Passman shows that for some sets X of integers, p-groups whose set of irreducible character degrees is precisely X have bounded nilpotence class, while for other choices of X, the nilpotence class is unbounded. This paper presents a theoren which shows some additional sets of character degrees which bound nilpotence class within the family of metabelian p-groups. In particular, it is shown that is the non-linear irreducible character degrees of G lie between pa and pb, where a ≤ b ≤ 2a − 2, then the nilpotence class of G is bounded by a function of p and b − a.

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18

LINH, CAO HUY. "Castelnuovo–Mumford regularity and Degree of nilpotency." Mathematical Proceedings of the Cambridge Philosophical Society 142, no. 3 (2007): 429–37. http://dx.doi.org/10.1017/s0305004106009819.

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AbstractIn this paper we show that the Castelnuovo–Mumford regularity of the associated graded module with respect to anm-primary idealIis effectively bounded by the degree of nilpotency ofI. From this it follows that there are only a finite number of Hilbert-Samuel functions for ideals with fixed degree of nilpotency.

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19

LOPATIN, ARTEM A., and IVAN P. SHESTAKOV. "ASSOCIATIVE NIL-ALGEBRAS OVER FINITE FIELDS." International Journal of Algebra and Computation 23, no. 08 (2013): 1881–94. http://dx.doi.org/10.1142/s0218196713500471.

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We study the nilpotency degree of a relatively free finitely generated associative algebra with the identity xn = 0 over a finite field 𝔽 with q elements. In the case of q ≥ n the nilpotency degree is proven to be the same as in the case of an infinite field of the same characteristic. In the case of q = n - 1 it is shown that the nilpotency degree differs from the nilpotency degree for an infinite field of the same characteristic by at most one. The nilpotency degree is explicitly computed for n = 3.

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20

Camina, Rachel D., Ainhoa Iñiguez, and Anitha Thillaisundaram. "Word problems for finite nilpotent groups." Archiv der Mathematik 115, no. 6 (2020): 599–609. http://dx.doi.org/10.1007/s00013-020-01504-w.

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AbstractLet w be a word in k variables. For a finite nilpotent group G, a conjecture of Amit states that $$N_w(1)\ge |G|^{k-1}$$ N w ( 1 ) ≥ | G | k - 1 , where for $$g\in G$$ g ∈ G , the quantity $$N_w(g)$$ N w ( g ) is the number of k-tuples $$(g_1,\ldots ,g_k)\in G^{(k)}$$ ( g 1 , … , g k ) ∈ G ( k ) such that $$w(g_1,\ldots ,g_k)={g}$$ w ( g 1 , … , g k ) = g . Currently, this conjecture is known to be true for groups of nilpotency class 2. Here we consider a generalized version of Amit’s conjecture, which states that $$N_w(g)\ge |G|^{k-1}$$ N w ( g ) ≥ | G | k - 1 for g a w-value in G, and prove that $$N_w(g)\ge |G|^{k-2}$$ N w ( g ) ≥ | G | k - 2 for finite groups G of odd order and nilpotency class 2. If w is a word in two variables, we further show that the generalized Amit conjecture holds for finite groups G of nilpotency class 2. In addition, we use character theory techniques to confirm the generalized Amit conjecture for finite p-groups (p a prime) with two distinct irreducible character degrees and a particular family of words. Finally, we discuss the related group properties of being rational and chiral, and show that every finite group of nilpotency class 2 is rational.

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21

Šnobl, L., and P. Winternitz. "All solvable extensions of a class of nilpotent Lie algebras of dimensionnand degree of nilpotencyn− 1." Journal of Physics A: Mathematical and Theoretical 42, no. 10 (2009): 105201. http://dx.doi.org/10.1088/1751-8113/42/10/105201.

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22

Avrunin, George S., and Jon F. Carlson. "Nilpotency degree of cohomology rings in characteristic two." Proceedings of the American Mathematical Society 118, no. 2 (1993): 339. http://dx.doi.org/10.1090/s0002-9939-1993-1129871-7.

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23

Minh, Pham Anh. "Nilpotency degree of cohomology rings in characteristic 3." Proceedings of the American Mathematical Society 130, no. 2 (2001): 307–10. http://dx.doi.org/10.1090/s0002-9939-01-06036-1.

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24

Minh, Pham Anh. "Nilpotency degree of cohomology rings in characteristic $p$." Proceedings of the American Mathematical Society 131, no. 2 (2002): 363–68. http://dx.doi.org/10.1090/s0002-9939-02-06550-4.

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25

ADAN-BANTE, EDITH. "CHARACTERS OF PRIME DEGREE." Glasgow Mathematical Journal 53, no. 3 (2011): 419–26. http://dx.doi.org/10.1017/s0017089511000413.

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AbstractLet G be a finite nilpotent group, χ and ψ be irreducible complex characters of G with prime degree. Assume that χ(1) = p. Then, either the product χψ is a multiple of an irreducible character or χψ is the linear combination of at least $\frac{p+1}{2}$ distinct irreducible characters.

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26

Lewis, Mark L. "Variations on average character degrees and p-nilpotence." Israel Journal of Mathematics 215, no. 2 (2016): 749–64. http://dx.doi.org/10.1007/s11856-016-1393-7.

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27

Tylyshchak, O. "On irreducibility of monomial 7 × 7−matrix over local ring." Bulletin of Taras Shevchenko National University of Kyiv. Series: Physics and Mathematics, no. 3 (2018): 37–44. http://dx.doi.org/10.17721/1812-5409.2018/3.5.

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We consider a monomial n × n-matrix, which corresponds to a cyclic permutation of the length n, over a commutative local principle ideals ring. Non-zero elements of a non-empty set of first columns of the matrix are identity element of the ring and non-zero elements of non-empty set of the rest columns are a fixed non-zero generator element of the Jacobson radical of the ring. It is known if number of identities or number of generator elements is exact 1 or if n < 7 and number of identities is relatively prime to n, then the matrix is irreducible. If the number of identities is not relatively prime to n, then the matrix is reducible. If the Jacobson radical of the ring is nilpotent of degree 2, then the 7 × 7-matrix of considered form with 3 or 4 identities is reducible. It has been shown that the 7 × 7-matrix is irreducible if the degree of nilpotency of the Jacobson radical of the ring is higher than 2. Some necessary conditions of reducibility of this square matrix of arbitrary size are also established.

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28

Minh, P. A. "Nilpotency degree of integral cohomology classes of p-groups." Archiv der Mathematik 79, no. 5 (2002): 328–34. http://dx.doi.org/10.1007/pl00012454.

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29

Dekimpe, Karel, та Daciberg Lima Gonçalves. "The 𝑅∞-property for nilpotent quotients of Baumslag–Solitar groups". Journal of Group Theory 23, № 3 (2020): 545–62. http://dx.doi.org/10.1515/jgth-2018-0182.

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AbstractA group G has the {R_{\infty}}-property if the number {R(\varphi)} of twisted conjugacy classes is infinite for any automorphism φ of G. For such a group G, the {R_{\infty}}-nilpotency degree is the least integer c such that {G/\gamma_{c+1}(G)} still has the {R_{\infty}}-property. In this paper, we determine the {R_{\infty}}-nilpotency degree of all Baumslag–Solitar groups.

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30

REZAEI, RASHID, and FRANCESCO G. RUSSO. "n-th relative nilpotency degree and relative n-isoclinism classes." Carpathian Journal of Mathematics 27, no. 1 (2011): 123–30. http://dx.doi.org/10.37193/cjm.2011.01.03.

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P. Hall introduced the notion of isoclinism between two groups more than 60 years ago. Successively, many authors have extended such a notion in different contexts. The present paper deals with the notion of relative n-isoclinism, given by N. S. Hekster in 1986, and with the notion of n-th relative nilpotency degree, recently introduced in literature.

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31

Huang, Wentao, Ting Chen, and Tianlong Gu. "Fourteen Limit Cycles in a Seven-Degree Nilpotent System." Abstract and Applied Analysis 2013 (2013): 1–5. http://dx.doi.org/10.1155/2013/398609.

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Center conditions and the bifurcation of limit cycles for a seven-degree polynomial differential system in which the origin is a nilpotent critical point are studied. Using the computer algebra system Mathematica, the first 14 quasi-Lyapunov constants of the origin are obtained, and then the conditions for the origin to be a center and the 14th-order fine focus are derived, respectively. Finally, we prove that the system has 14 limit cycles bifurcated from the origin under a small perturbation. As far as we know, this is the first example of a seven-degree system with 14 limit cycles bifurcated from a nilpotent critical point.

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32

Cagliero, Leandro, Fernando Levstein, and Fernando Szechtman. "Nilpotency degree of the nilradical of a solvable Lie algebra on two generators and uniserial modules associated to free nilpotent Lie algebras." Journal of Algebra 585 (November 2021): 447–83. http://dx.doi.org/10.1016/j.jalgebra.2021.06.008.

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Isaacs, I. M., and Alexander Moretó. "The Character Degrees and Nilpotence Class of a p-Group." Journal of Algebra 238, no. 2 (2001): 827–42. http://dx.doi.org/10.1006/jabr.2000.8651.

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Rezaei, Rashid, and Francesco G. Russo. "BOUNDS FOR THE RELATIVE n-TH NILPOTENCY DEGREE IN COMPACT GROUPS." Asian-European Journal of Mathematics 04, no. 03 (2011): 495–506. http://dx.doi.org/10.1142/s1793557111000411.

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The line of investigation of the present paper goes back to a classical work of W. H. Gustafson of the 1973, in which it is described the probability that two randomly chosen group elements commute. In the same work, he gave some bounds for this kind of probability, providing information on the group structure. We have recently obtained some generalizations of his results for finite groups. Here we improve them in the context of the compact groups.

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35

Lopatin, Artem A. "On the nilpotency degree of the algebra with identity xn=0." Journal of Algebra 371 (December 2012): 350–66. http://dx.doi.org/10.1016/j.jalgebra.2012.08.007.

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Rezaei, Rashid, and Ahmad Erfanian. "A NOTE ON THE RELATIVE COMMUTATIVITY DEGREE OF FINITE GROUPS." Asian-European Journal of Mathematics 07, no. 01 (2014): 1450017. http://dx.doi.org/10.1142/s179355711450017x.

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The purpose of this paper is to give a relation between the notion of the commutativity degree of a finite group G (denoted by d(G)) and that of isoclinism between G and an extra special p-group, where p is the smallest prime number dividing |G|. Moreover, some improvements of the results on the relative commutativity degree and relative n th nilpotency degree of a subgroup of finite groups given in [A. Erfanian, R. Rezaei and P. Lescot, On the relative commutativity degree of a subgroup of a finite group, Comm. Algebra35 (2007) 4183–4197] are also stated in this paper.

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Espuelas, A. "Character Degrees of Nilpotent-by-Metacyclic Groups." Journal of Algebra 162, no. 2 (1993): 531–34. http://dx.doi.org/10.1006/jabr.1993.1268.

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Iranmanesh, A., and M. Viseh. "A Remark on Character Degrees and Nilpotence Class in $p$-Groups." Missouri Journal of Mathematical Sciences 19, no. 1 (2007): 49–51. http://dx.doi.org/10.35834/mjms/1316092237.

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Bello, Muhammed, Nor Muhainiah Mohd Ali, and Nurfarah Zulkifli. "A Systematic Approach to Group Properties Using its Geometric Structure." European Journal of Pure and Applied Mathematics 13, no. 1 (2020): 84–95. http://dx.doi.org/10.29020/nybg.ejpam.v13i1.3587.

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The algebraic properties of a group can be explored through the relationship among its elements. In this paper, we define the graph that establishes a systematic relationship among the group elements. Let G be a finite group, the order product prime graph of a group G, is a graph having the elements of G as its vertices and two vertices are adjacent if and only if the product of their order is a prime power. We give the general presentation for the graph on dihedral groups and cyclic groups and classify finite dihedral groups and cyclic groups in terms of the order product prime graphs as one of connected, complete, regular and planar. We also obtained some invariants of the graph such as its diameter, girth,independent number and the clique number. Furthermore, we used thevertex-cut of the graph in determining the nilpotency status of dihedralgroups. The graph on dihedral groups is proven to be regular and complete only if the degree of the corresponding group is even prime power and connected for all prime power degree. It is also proven on cyclic groups to be both regular, complete and connected if the group has prime power order. Additionally, the result turn out to show that any dihedral group whose order product prime graph’s vertex-cut is greater than one is nilpotent. We also show that the order product prime graph is planar only when the degree of the group is three for dihedral groups and less than five for cyclic groups. Our final result shows that the order product prime graphs of any two isomorphic groups are isomophic.

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40

Bello, Muhammed, Nor Muhainiah Mohd Ali, and Nurfarah Zulkifli. "A Systematic Approach to Group Properties Using its Geometric Structure." European Journal of Pure and Applied Mathematics 13, no. 1 (2020): 84–95. http://dx.doi.org/10.29020/nybg.ejpam.v1i1.3587.

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The algebraic properties of a group can be explored through the relationship among its elements. In this paper, we define the graph that establishes a systematic relationship among the group elements. Let G be a finite group, the order product prime graph of a group G, is a graph having the elements of G as its vertices and two vertices are adjacent if and only if the product of their order is a prime power. We give the general presentation for the graph on dihedral groups and cyclic groups and classify finite dihedral groups and cyclic groups in terms of the order product prime graphs as one of connected, complete, regular and planar. We also obtained some invariants of the graph such as its diameter, girth,independent number and the clique number. Furthermore, we used thevertex-cut of the graph in determining the nilpotency status of dihedralgroups. The graph on dihedral groups is proven to be regular and complete only if the degree of the corresponding group is even prime power and connected for all prime power degree. It is also proven on cyclic groups to be both regular, complete and connected if the group has prime power order. Additionally, the result turn out to show that any dihedral group whose order product prime graph’s vertex-cut is greater than one is nilpotent. We also show that the order product prime graph is planar only when the degree of the group is three for dihedral groups and less than five for cyclic groups. Our final result shows that the order product prime graphs of any two isomorphic groups are isomophic.

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41

Adyan, S. I., and N. N. Repin. "Exponential lower estimate of the degree of nilpotency of Engel Lie algebras." Mathematical Notes of the Academy of Sciences of the USSR 39, no. 3 (1986): 244–49. http://dx.doi.org/10.1007/bf01170256.

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42

Corbera, Montserrat, and Claudia Valls. "Reversible polynomial Hamiltonian systems of degree 3 with nilpotent saddles." Discrete & Continuous Dynamical Systems - B 22, no. 11 (2017): 0. http://dx.doi.org/10.3934/dcdsb.2020225.

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43

Dias, Fabio Scalco, Jaume Llibre, and Claudia Valls. "Polynomial Hamiltonian systems of degree 3 with symmetric nilpotent centers." Mathematics and Computers in Simulation 144 (February 2018): 60–77. http://dx.doi.org/10.1016/j.matcom.2017.06.002.

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44

Silva, Adriana R., and Israel Vainsencher. "Degree of the variety of pairs of nilpotent commuting matrices." Bulletin of the Brazilian Mathematical Society, New Series 45, no. 4 (2014): 837–63. http://dx.doi.org/10.1007/s00574-014-0078-2.

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45

Zhou, Feng, and Heguo Liu. "A nilpotency criterion related to the minimal degree of a nonlinear irreducible character." Journal of Algebra and Its Applications 16, no. 06 (2017): 1750102. http://dx.doi.org/10.1142/s021949881750102x.

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Let [Formula: see text] be a finite nonabelian group, let [Formula: see text] be the minimal degree of a nonlinear irreducible character of [Formula: see text] and suppose that [Formula: see text] for some positive integer [Formula: see text]. Then [Formula: see text] is nilpotent.

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46

Navarro, Gabriel, and Noelia Rizo. "Nilpotent and perfect groups with the same set of character degrees." Journal of Algebra and Its Applications 13, no. 08 (2014): 1450061. http://dx.doi.org/10.1142/s0219498814500613.

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47

Pálfy, P. P. "On the character degree graph of solvable groups. II. Discon- nected graphs." Studia Scientiarum Mathematicarum Hungarica 38, no. 1-4 (2001): 339–55. http://dx.doi.org/10.1556/sscmath.38.2001.1-4.25.

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Let G be a finite solvable group and ¡ a set of primes such that the degree of each irreducible character of G is either a ¡-number or a ¡0-number. We show that such groups have a very restricted structure, for example, their nilpotent length is at most 4. We also prove that Huppert's ^{ÿ Conjecture is valid for these groups.

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Pang, Linna, and Jiakuan Lu. "Finite groups and degrees of irreducible monomial characters II." Journal of Algebra and Its Applications 16, no. 12 (2017): 1750231. http://dx.doi.org/10.1142/s0219498817502310.

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Let [Formula: see text] be a finite solvable group, and let [Formula: see text] be a prime. We obtain some conditions for [Formula: see text] to be [Formula: see text]-nilpotent or [Formula: see text]-closed in terms of irreducible monomial characters.

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49

Grishkov, Alexander, and Said Sidki. "Representing Idempotents as a Sum of Two Nilpotents of Degree Four." Communications in Algebra 32, no. 2 (2004): 715–26. http://dx.doi.org/10.1081/agb-120027925.

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Erfanian, Ahmad, Rashid Rezaei, and Francesco Russo. "Relative n-isoclinism classes and relative n-th nilpotency degree of finite groups." Filomat 27, no. 2 (2013): 365–69. http://dx.doi.org/10.2298/fil1302365e.

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