In this paper, we consider the finitely 2-generated groups $K(s,l)$ and $G_m$ as follows:$$K(s,l)=langle a,b|ab^s=b^la, ba^s=a^lbrangle,\G_m=langle a,b|a^m=b^m=1, {[a,b]}^a=[a,b], {[a,b]}^b=[a,b]rangle$$ and find the explicit formulas for the probability of having nth-roots for them. Also, we investigate integers n for which, these groups are n-central.

We show deterministic polynomial time algorithms over some family of finite fields for solving univariate polynomial equations and some related problems such as taking nth roots, constructing nth nonresidues, constructing primitive elements and computing elliptic curve “nth roots”. In additional, we present a deterministic polynomial time primality test for some family of integers. All algorith...

We describe the image of the absolute Galois group acting on the pro-l completion of the fundamental group of the Gm minus Nth roots of unity. We relate the structure of the image with geometry and topology of modular varieties for the congruence subgroups 01(m; N ) of GLm(Z) for m = 1, 2, 3 . . . .

in this paper, we consider the finitely 2-generated groups k(s,l) and g_m as follows:k(s,l) = ;g_m = and find the explicit formulas for the probability of having nth-roots for them. also weinvestigate integers n for which, these groups are n-central.

here we consider all finite non-abelian 2-generator $p$-groups ($p$ an odd prime) of nilpotency class two and study the probability of having $n^{th}$-roots of them. also we find integers $n$ for which, these groups are $n$-central.

Based on the generalized continued-fraction method for finding the nth roots of real numbers, this paper presents a fast computation method for finding the principal nth roots of complex matrices. Computation algorithms with high convergence rates are developed, and their global convergence properties are investigated from the viewpoint of systems theory.