A regular map M is a cellular decomposition of a surface such that its automorphism group Aut(M) acts transitively on the flags of M. It can be shown that if a Sylow subgroup P ≤ Aut(M) has order coprime to the Euler characteristic of the supporting surface, then P is cyclic or dihedral. This observation motivates the topic of the current paper, where we study regular maps whose automorphism gr...

Let p be a prime,G a finite group which has a normal p-subgroup containing its own centralizer inG, and R a commutative local ring with residue class field of characteristic p. In this paper, it is shown that if α is an augmented automorphism of RG which fixes a Sylow p-subgroup P of G, there is ρ ∈ Aut(G) such that xαρ = x for all x ∈ P , and αρ is an inner automorphism of RG.

‎let $g$ be a finite group and $pi_{e}(g)$ be the set of element‎ ‎orders of $g$‎. ‎let $k in pi_{e}(g)$ and $m_{k}$ be the number of‎ ‎elements of order $k$ in $g$‎. ‎set nse($g$):=${ m_{k} | k in‎ ‎pi_{e}(g)}$‎. ‎in this paper‎, ‎we prove that if $g$ is a group such‎ ‎that nse($g$)=nse($psl(2‎, ‎25)$)‎, ‎then $g cong psl(2‎, ‎25) $‎.

‎let $g$ be a finite group and $pi_{e}(g)$ be the set of element‎ ‎orders of $g$‎. ‎let $k in pi_{e}(g)$ and $m_{k}$ be the number of‎ ‎elements of order $k$ in $g$‎. ‎set nse($g$):=${ m_{k} | k in‎ ‎pi_{e}(g)}$‎. ‎in this paper‎, ‎we prove that if $g$ is a group such‎ ‎that nse($g$)=nse($psl(2‎, ‎25)$)‎, ‎then $g cong psl(2‎, ‎25) $‎.