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### Page 25 of Mathematical Reviews Vol. , Issue 2004a [page]

2004 Mathematical Reviews
Let G be a finite p-group (p a prime) of exponent p and nilpotent class 4, and consider a 3-step Fibonacci sequence r = (r;) in G, with some initial data ro, r; and rz.  ...  a finite nilpotent group.  ...

### Page 3076 of Mathematical Reviews Vol. , Issue 99e [page]

1999 Mathematical Reviews
Soc. (2) 49 (1994), no. 1, 83-92; MR 94k:20059] showed that the lengths of ordinary 2-step Fibonacci sequences are equal to the length of the 2-step Fibonacci recurrences in finite nilpotent groups of  ...  Fibonacci Quart. 36 (1998), no. 3, 216-221. The study of Fibonacci sequences in groups began with work of D. D. Wall [Amer. Math.  ...

### Page 4218 of Mathematical Reviews Vol. , Issue 82j [page]

1982 Mathematical Reviews
In Section 3 there is a gap in the discussion of equations over cyclic groups. The example given there illustrates a new application of small cancellation theory.  ...  He proves (Theorem 1) that they are then finite extensions of a subgroup B whose centre is of finite index, such that for every element in the group but not in B the centralizer in the whole group is finite  ...

### Page 5903 of Mathematical Reviews Vol. , Issue 96j [page]

1996 Mathematical Reviews
Any noncyclic relatively free group of finite rank in a variety  ...  They use the famous Tits theorem in proving that a relatively free group that is not nilpotent-by-abelian-by-finite has automorphism group that is not linear over a field of characteristic zero.  ...

### Page 2726 of Mathematical Reviews Vol. , Issue 96e [page]

1996 Mathematical Reviews
The author defines a sequence of groups (k) "ae ee F,*) = (a1,°++,@n| @i@j,) =@i42, 1=1,2,---,m) which are generalisations of the Fibonacci groups (the case k = 1).  ...  It follows as a corollary that G is finite-by-nilpotent if and only if the Frattini factor group G/®(G) is finite-by-nilpotent. H.  ...

### Infinite systems of linear equations and mathematical logic

O.M. Tabachnikova
2002 Linear Algebra and its Applications
We will demonstrate a result in linear algebra which is a consequence of a now classical result of mathematical logic, and then illustrate an application of this result to group theory.  ...  It was demonstrated in  that the fundamental period of a Fibonacci sequence in a finite p-group of exponent p > 3 and nilpotency class at most 4 is a divisor of the fundamental period of the usual  ...  Fibonacci sequence in the integers modulo p.  ...

### Page 6566 of Mathematical Reviews Vol. , Issue 2003i [page]

2003 Mathematical Reviews
In this paper the main result is the following. Theorem: Let G be a nontrivial finite p-group of exponent p and nilpotency class 2. Then we have k(G) =k.  ...  2003i: 1 1025 2003i:11025 11B39 20D15 Dikici, Ramazan (TR-ATAED-MD; Erzurum); Ozkan, Engin (TR-ATAS; Erzurum) An application of Fibonacci sequences in groups. (English summary) Appl. Math.  ...

### Asymptotically optimal k-step nilpotency of quadratic algebras and the Fibonacci numbers [article]

Natalia Iyudu, Stanislav Shkarin
2016 arXiv   pre-print
It follows from the Golod--Shafarevich theorem that if R is an associative algebra given by n generators and d<n^2/4cos^-2(π/k+1) quadratic relations, then R is not k-step nilpotent.  ...  We show that the above estimate is asymptotically optimal, and establish number of related results. For example, we show that for any k this estimate is attained for ifinitely many n.  ...  Acknowledgements We are grateful to the Max-Planck-Institute for Mathematics in Bonn and to IHES, where parts of this research have been done, for hospitality, support, and excellent research atmosphere  ...

### Page 5798 of Mathematical Reviews Vol. , Issue 2001H [page]

2001 Mathematical Reviews
| n=1,2,3,---} and Fibonacci sequences can be generated in real time by cellular automata with 1-bit inter-cell communications.  ...  It is shown that an algebra (A,z) is nilpotent iff it is a ho- momorphic image of a subalgebra of a quasi-cascade-product of unary nilpotent algebras; (A, ) is definite iff it is a homomorphic image of  ...

### Relatively Free Semigroups of Intermediate Growth

L.M Shneerson
2001 Journal of Algebra
He proved that every f.g. group of polynomial growth contains a nilpotent subgroup of w x finite index.  ...  Sapir proved w x 23 that the variety of all nilpotent groups of class c cannot be defined by the system of identities over a finite set of variables even in the class of monoids for c G 2.  ...  The equality 160 shows that induces a one-to-one correspondence Ä 4 between the set of all finite sequences ␣ as above and the set of all n words of the type ⌬ U over the alphabet ⌽.  ...

### On the Basick-nacci Sequences in Finite Groups

2011 Discrete Dynamics in Nature and Society
We define the basick-nacci sequences and the basic periods of these sequences in finite groups, then we obtain the basic periods of the basick-nacci sequences and the periods of thek-nacci sequences in  ...  symmetric groupS4, its subgroups, and binary polyhedral groups which related with these groups.  ...  Acknowledgments The authors thank the referees for their valuable suggestions which improved the presentation of the paper.  ...

### Page 2659 of Mathematical Reviews Vol. , Issue 93e [page]

1993 Mathematical Reviews
In Section 2, it is shown that an elementary nilpo- tent Lie group can act freely and distally on some nilmanifold. A nilmanifold is a quotient of a nilpotent Lie group by a uniform lattice.  ...  Contents: Alejandro Adem, On the geometry and cohomology of finite simple groups (1-9); D. J. Benson [David John Benson], Resolutions and Poincaré duality for finite groups (10-19); A. J.  ...

### The Metallic Ratio of Pulsating Fibonacci Sequences

Kittipong Laipaporn, Kiattiyot Phibul, Prathomjit Khachorncharoenkul
2022 Symmetry
In this paper, we investigate the metallic ratio—a generalized version of the golden ratio—of pulsating Fibonacci sequences in three forms.  ...  The third form is the Fibonacci sequence in bipolar quantum linear algebra (BQLA), which also pulsates.  ...  We would like to express our appreciation to Rasimate Maungchang for triggering the golden ratio issue during our research on the pulsating pm, cq-Fibonacci sequence.  ...

### Page 3818 of Mathematical Reviews Vol. , Issue 95g [page]

1995 Mathematical Reviews
Abbott (3-AB; Edmonton, AB) 95g:11009 11B39 20D15 Aydin, Huseyin (TR-ATAS; Erzurum); Smith, Geoff C. (4-BATH-SM; Bath) Fourier analysis in finite nilpotent groups.  ...  Applications of Fibonacci numbers, Vol. 5 (St. Andrews, 1992), 49-59, Kluwer Acad. Publ., Dordrecht, 1993. Let s = (s;), so =0, s; = 1, denote the ordinary Fibonacci se- quence, modulo n.  ...

### Page 6213 of Mathematical Reviews Vol. , Issue 94k [page]

1994 Mathematical Reviews
Smith, Fourier analysis in finite nilpotent groups (49-59); Huseyin Aydin, Ramazan Dikici and Geoff C.  ...  Peter Gumm (D-MRBG; Marburg) See also 20070. 11 NUMBER THEORY 94k:11001 11-06 11B39 ¥* Applications of Fibonacci numbers. Vol. 5.  ...
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