### Periodicity invariant of finitely generated algebraic structures

#### Abstract

sources in the theory of invariants of finite groups and it forms an interesting and relatively self-contained nook in the

imposing edifice of group theory. One of the deepest and important results of the related theory of finite

groups is a complete classification of all periodic groups, that is, the finite groups with periodic properties. If an integer

be $k\geq 2$, let $S$ will be a finite $k$-generated as well as non-associative algebraic structure $S=<A>$, where

$A=\lbrace a_{1}, a_{2},\dots, a_{k}\rbrace$, and the sequence

$$x_{i}=\left\{

\begin{array}{ll}

a_{i}, & 1\leq i\leq k, \\

x_{i-k}(x_{i-k+1}(\ldots(x_{i-3}(x_{i-2}x_{i-1}))\ldots)), & i>k,

\end{array}

\right.

$$

is called the $k$-nacci sequence of $S$ with respect to the generating set $A$, as denoted in $k_{A}(S)$.

When $k_{A}(S)$ is periodic, we will use the length of the period of the periodicity length of $S$ proportional to $A$ in $LEN_{A}(S)$

and the minimum of the positive integers of $LEN_A(S)$ will be mentioned as periodicity invariant of $S$, denoted in $\lambda_k(S)$. However, this invariant has

been studied for groups and semigroups during the years as well as the associative property of $S$ where above sequence was reduced to

$x_i=x_{i-k}x_{i-k+1}\dots x_{i-3}x_{i-2}x_{i-1}$, for every $i\geq k+1$. Thus, we attempt to give explicit upper

bounds for the periodicity invariant of two infinite classes of

finite non-associative $3$-generated algebraic structures. Moreover, two classes of non-isomorphic Moufang loops of the same periodicity length were obtained in the study.

#### Keywords

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