Let G be a finite group and pe (G) be the set of element orders of G. Let kÎpe (G) and m k be the number of elements of order k in G. Set nse (G): = {mk çk Îpe (G)} g. In this paper, we prove that if G is a group such that nse (G)=nse (PSL (2; 25)), then G@PSL (2; 25).

We exhibit presentations of the Von Dyck groups D(2,3,m),m ,3, in terms of two generators of order m satisfying three relations, one of which is Artin’, s braid relation. By dropping the relation which fi, xes the order of the generators we obtain the universal covering groups of the corresponding Von Dyck groups. In the cases m = 3,4,5, these are respectively the double covers of the fi, nite rotational tetrahedral, octahedral and icosahedral groups. When m ,6 we obtain infi, nite covers of the corresponding infi, nite Von Dyck groups. The interesting cases arise for m ,7 when these groups act as discrete groups of isometries of the hyperbolic plane. Imposing a suitable third relation we obtain three-relator presentations of PSL(2,m). We discover two general formulas presenting these as factors of D(2,3,m). The fi, rst one works for any odd m and is essentially equivalent to the shortest known presentation of Sunday [J. Sunday, Presentations of the groups SL(2,m) and PSL(2,m), Canadian J. Math., 24 (1972) 1129–, 1131]. The second applies to the cases m ,, 2 (mod 3), m , = 11(mod 30), and is substantively shorter. Additionally, by random search, we fi, nd many effi, cient presentations of fi, nite simple Chevalley groups PSL(2,q) as factors of D(2,3,m) where m divides the order of the group. The only other simple group that we found in this way is the sporadic Janko group J2.

Let G be a finite group. We denote by ψ(G) the integer ågÎGo(g), where o(g) denotes the order of gÎG. Here we show that ψ(A5)<ψ(G) for every non-simple group G of order 60, where A5 is the alternating group of degree 5. Also we prove that ψ(PSL (2,7))<ψ(G) for all non-simple groups G of order 168. These two results confirm the conjecture posed in [J. Algebra Appl., 10 No.2 (2011) 187-190] for simple groups A5 and PSL (2,7).

This qualitative study investigates the process of learning verbs, as one of the fundamental aspects of learning Persian language, by PSL (Persian as a second language) Learners through dynamic assessment and using new technologies which are in contradiction with the traditional approaches of teaching and assessment. In so doing, two Iraqi students of Razi University were subjected to a four-month enrichment program that captured their microgenetic development and involved using IMO which, as a realization of new technologies, provides its users with the opportunity of synchronized electronic communication. Results indicate that each learner upon receiving the right level of instruction and prompts moved along the continuum of mediation internalization to get closer to the self-regulation extreme which in turn denotes their social and cognitive development in their Zone of Proximal Development in using the target structures. That is, dynamic assessment via electronic communication equipped the researchers with an in-depth knowledge of each learner`s position along the continuum of other-self regulation which was followed by the appropriate level of instruction and led to his development. The findings of this study contribute to the area of second language teaching in general and Persian language teaching in particular.

A , nite group G is said to be (l,m,n)-generated, if it is a quotient group of the triangle group T(l,m,n) = x,y,zjxl = ym = zn = xyz = 1 , : In [J. Moori, (p,q,r)-generations for the Janko groups J1 and J2, Nova J. Algebra and Geometry, 2 (1993), no. 3, 277{285], Moori posed the question of , nding all the (p,q,r) triples, where p,q and r are prime numbers, such that a non-abelian , nite simple group G is (p,q,r)-generated. Also for a , nite simple group G and a con-jugacy class X of G,the rank of X in G is de, ned to be the minimal number of elements of X generating G: In this paper we investigate these two generational prob-lems for the group PSL(3,7),where we will determine the (p,q,r)-generations and the ranks of the classes of PSL(3,7): We approach these kind of generations using the structure constant method. GAP [The GAP Group, GAP { Groups, Algorithms, and Programming, Version 4. 9. 3,2018. (http: //www. gap-system. org)] is used in our computations.

Let p = q4+q3+q2+q+1/(5; q-1) be a prime number, where q is a prime power. In this paper, we will show G  PSL(5, q) if and only if |G| = |PSL(5; q)|, and G has a conjugacy class size |PSL(5; q)|/p. Further, the validity of a conjecture of J. G. Thompson is generalized to the groups under consideration by a new way.

If F is a subfield of C, then a square matrix over F with non-negative integral trace is called a quasi-permutation matrix over F. For a finite group G, let p(G) denote the minimal degree of a faithful permutation representation of G and let q(G) and c(G) denote the minimal degree of a faithful representation of G by quasi-permutation matrices over the rationals and the complex numbers, respectively. Finally r(G) denotes the minimal degree of a faithful rational valued complex character of G. In this paper, p(G), q(G), c(G) and r(G) are calculated for the groups SL(3,q) and PSL (3,q).

In this paper, using a method of construction of 1-designs which are not necessarily symmetric, introduced by Key and Moori, we determine a number of 1-designs with interesting parameters from the maximal subgroups and the conjugacy classes of elements of the group PSL(2,q) for q a power of an odd prime.