The aim of this paper is basically to study some of the standard properties of the intuitionistic fuzzy subgroups under a crisp map. Also, we study some properties of intuitionistic fuzzy normal subgroups.

for a finite group $g$ let $nu(g)$ denote the number of conjugacy classes of non-normal subgroups of $g$. we give a short proof of a theorem of brandl, which classifies finite groups with $nu(g)=1$.

let $g$ be a finite group‎. ‎a subgroup‎ ‎$h$ of $g$ is called an $mathcal{h}$-subgroup in‎ ‎$g$ if $n_g(h)cap h^{g}leq h$ for all $gin‎ ‎g$. a subgroup $h$ of $g$ is called a weakly‎ ‎$mathcal{h}^{ast}$-subgroup in $g$ if there exists a‎ ‎subgroup $k$ of $g$ such that $g=hk$ and $hcap‎ ‎k$ is an $mathcal{h}$-subgroup in $g$. we‎ ‎investigate the structure of the finite group $g$ under the‎ ‎as...

We study homogeneous geodesics of sub-Riemannian manifolds, i.e. , normal that are orbits one-parametric subgroups isometries. obtain a criterion for geodesic to be in terms its initial momentum. prove any weakly commutative space is orbit, means all homogeneous. discuss some examples orbit manifolds. In particular, we show Carnot groups only step 1 and 2. Finally, get broad condition existence...

In this paper, the notions of   , q  -Q-fuzzy subgroup and normal subgroup are introduced and some of their properties are investigated.