For a given integer n = p α 1

We study 2-step general Fibonacci sequences in the generalized quaternion groupsQ4n. In cases where the sequences are proved to be simply periodic, we obtain the periods of 2-step general Fibonacci sequences.

FIBONACCI LENGTHS OF ALL FINITE p-GROUPS OF EXPONENT p 2 ДОВЖИНИ ФIБОНАЧЧI ДЛЯ ВСIХ СКIНЧЕННИХ p-ГРУП ЕКСПОНЕНТИ p 2 The Fibonacci lengths of finite p-groups were studied by Dikici and co-authors since 1992. All of the considered groups are of exponent p, and the lengths depend on the Wall number k(p). The p-groups of nilpotency class 3 and exponent p were studied in 2004 also by Dikici. In the present paper, we study all p-groups of nilpotency class 3 and exponent p 2. We thus complete the study of Fibonacci lengths of all p-groups of order p 4 , proving that the Fibonacci length is k(p 2). Довжини Фiбоначчi скiнченних p-груп вивчалися Дiкiчi та спiвавторами з 1992 року. Всi групи, що розглядалися, були групами експоненти p, а всi довжини залежали вiд числа Уолла k(p). p-Групи класу нiльпотентностi 3 i експоненти p були також дослiдженi Дiкiчi у 2004 роцi. У данiй статтi ми вивчаємо всi p-групи класу нiльпотентностi 3 i експоненти p 2. Цим завершується дослiдження довжини Фiбоначчi всiх p-груп порядку p 4 ; при цьому доведено, що довжина Фiбоначчi дорiвнює k(p 2).

For a finite algebraic structure A the commutativity degree of A, denoted by P r (A), is the probability that two elements of A commute. Since 1973, P r (A) studied for finite groups showing that P r (A) ≤ 5 8 . The study of "almost commutative" semigroups in 2011 showed that for some finite semigroup A, P r (A) > 5 8 and even P r (A) may be arbitrarily close to 1. Since 0 ≤ P r (A) ≤ 1 then looking for algebraic structures A such that P r (A) → 0, is of interest, for, the centralizer of every element of such A should be a singleton. In this paper for every integer n ≥ 2 and every prime p ≥ 3 we give an infinite class of finite semigroup rings A n,p = Z p (P n ) of order p n where, P n is a non-commutative semigroup and show that P r (A n,p ) → 0, for sufficiently large values of n and p. We name such semigroup rings as "extremely non-commutative" semigroup rings.