the aim of this paper is to study the $(alpha‎, ‎gamma)$-prolongation of central extensions‎. ‎we obtain the obstruction theory for $(alpha‎, ‎gamma)$-prolongations and classify $(alpha‎, ‎gamma)$-prolongations thanks to low-dimensional cohomology groups of groups‎.

let $f_q d_{2n}$ be the group algebra of $d_{2n}$, the dihedral group of order $2n$ over $f_q=gf(q)$. in this paper, we establish the structure of $u(f_{2^k}d_{2n})$, the unit group of $f_{2^k}d_{2n}$ and that of its normalized unitary subgroup $v_*(f_{2^k}d_{2n})$ with respect to canonical involution $*$ when $n$ is odd.

let $g$ be a $p$-group of order $p^n$ and $phi$=$phi(g)$ be the frattini subgroup of $g$. it is shown that the nilpotency class of $autf(g)$, the group of all automorphisms of $g$ centralizing $g/ fr(g)$, takes the maximum value $n-2$ if and only if $g$ is of maximal class. we also determine the nilpotency class of $autf(g)$ when $g$ is a finite abelian $p$-group.

in this paper, we study fuzzy substructures in connection withhv-structures. the original idea comes from geometry, especially from thetwo dimensional euclidean vector space. using parameters, we obtain a largenumber of hyperstructures of the group-like or ring-like types. we connect,also, the mentioned hyperstructures with the theta-operations to obtain morestrict hyperstructures, as hv-groups...

in this paper, we verify the solvability of free product of finite cyclic groups with topological methods. we use cayley graphs and everitt methods to construct suitable 2-complexes corresponding to the presentations of groups and their commutator subgroups. in particular, using these methods, we prove that the commutator subgroup of $mathbb{z}_{m}*mathbb{z}_{n}$ is free of rank $(m-1)(n-1)$ fo...