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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