Let G be a finite group which is not a cyclic p-group, p a prime number. We define an undirected simple graph ∆(G) whose vertices are the proper subgroups of G, which are not contained in the Frattini subgroup of G and two vertices H and K are joined by an edge if and only if G = ⟨H,K⟩. In this paper we classify finite groups with planar graph. Our result shows that only few groups have planar ...

a longstanding conjecture asserts that every finite nonabelian $p$-group admits a noninner automorphism of order $p$‎. ‎let $g$ be a finite nonabelian $p$-group‎. ‎it is known that if $g$ is regular or of nilpotency class $2$ or the commutator subgroup of $g$ is cyclic‎, ‎or $g/z(g)$ is powerful‎, ‎then $g$ has a noninner automorphism of order $p$ leaving either the center $z(g)$ or the frattin...

a longstanding conjecture asserts that every finite nonabelian $p$-group admits a noninner automorphism of order $p$. let $g$ be a finite nonabelian $p$-group. it is known that if $g$ is regular or of nilpotency class $2$ or the commutator subgroup of $g$ is cyclic, or $g/z(g)$ is powerful, then $g$ has a noninner automorphism of order $p$ leaving either the center $z(g)$ or the frattini subgro...

a normal subgroup $n$ of a group $g$ is said to be an‎ ‎$emph{omissible}$ subgroup of $g$ if it has the following property‎: ‎whenever $xleq g$ is such that $g=xn$‎, ‎then $g=x$‎. ‎in this note we construct various groups $g$‎, ‎each of which has an omissible subgroup $nneq 1$ such that $g/ncong sl_2(k)$ where $k$ is a field of positive characteristic‎.

Let G = A HB be the generalized free product of the groups A and B with the amalgamated subgroup H. Also, let λ(G) and ψ(G) represent the lower near Frattini subgroup and the near Frattini subgroup of G, respectively. If G is finitely generated and residually finite, then we show that ψ(G) ≤ H, provided H satisfies a nontrivial identical relation. Also, we prove that if G is residually finite, ...

For a finite group G and an arbitrary prime p, let S (G) denote the P intersection of all maximal subgroups M of G such that [G:M] is both composite and not divisible by p; if no such M exists we set S (G) G. Some properties of P G are considered involving S (G). In particular, we obtain a characterization of P G when each M in the definition of S (G) is nilpotent. P

We define the toplogical generating rank dtop(G) of a topological group G as the minimal number of elements of G which generate a dense subgroup of G. I will present joint work with Gena Noskov, partly in progress, about our attempts to compute the topological generating rank for connected Lie groups. The cases that G is semisimple, where dtop(G) is slightly bigger than one, and that G is nilpo...