abstractlet w be a non-empty subset of a free group. the automorphism of a group g is said to be a marginal automorphism, if for all x in g,x^−1alpha(x) in w^*(g), where w^*(g) is the marginal subgroup of g.in this paper, we give necessary and sufficient condition for a purelynon-abelian p-group g, such that the set of all marginal automorphismsof g forms an elementary abelian p-group.

Let p be a prime. In this note we make explicit some results on the canonical subgroup of an elliptic curve E over the ring of integers Rp of Cp implicit in [K-pPMF]. In particular, if ω generates Ω1E/Rp and E has a canonical subgroup CE , knowlege of the Hasse invariant of the reduction of (E,ω) modulo p is equivalent to knowledge of the pair (CE , ω|CE). 1. Group Schemes of order p. Let μ den...

in this paper, we extend the construction of a fuzzy subgroup generated by a fuzzy subset to $l$-setting. this construction is illustrated by an example. we also prove that for an $l$-subset of a group, the subgroup generated by its level subset is the level subset of the subgroup generated by that $l$-subset provided the given $l$-subset possesses sup-property.

Let (W,S) be a Coxeter system, and let X be a subset of S. The subgroup of W generated by X is denoted by WX and is called a parabolic subgroup. We give the precise definition of the commensurator of a subgroup in a group. In particular, the commensurator of WX in W is the subgroup of w in W such that wWXw ∩WX has finite index in both WX and wWXw . The subgroup WX can be decomposed in the form ...

let $g$ be a finite group‎. ‎a subgroup‎ ‎$h$ of $g$ is called an $mathcal h $ -subgroup in‎ ‎$g$ if $n_g (h)cap h^gleq h$ 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 ‎assump...

Given a toric affine algebraic variety $X$ and collection of one-parameter unipotent subgroups $U_1,\ldots,U_s$ $\mathop{\rm Aut}(X)$ which are normalized by the torus acting on $X$, we show that group $G$ generated verifies following alternative Tits' type: either is group, or it contains non-abelian free subgroup. We deduce if $2$-transitive $G$-orbit in then subgroup, so, exponential growth.

We conclude thhat T is a subgroup. Now we exhibit an example where T fails to be a subgroup when G is non-abelian. Let G = 〈x, y|x = y = i 〉. Observe that the product xy has infinite order since the product (xy)(xy) is no longer reducible because we lack the commutativity of x with y. Hence, x, y ∈ T but (xy) / ∈ T . Exercise (7). Fix some n ∈ Z with n > 1. Find the torsion subgroup of Z×(Z/nZ)...

The subgroups of DN are either cyclic or dihedral. The possible cyclic subgroups are of the form 〈(x, 0)〉 where x ∈ ZN is either 0 or some divisor of N . The possible dihedral subgroups are of the form 〈(y, 1)〉 where y ∈ ZN , and of the form 〈(x, 0), (y, 1)〉 where x ∈ ZN is some divisor of N and y ∈ Zx. A result of Ettinger and Høyer reduces the general dihedral HSP, in which the hidden subgrou...

a subgroup $h$ is said to be $nc$-supplemented in a group $g$ if there exists a subgroup $kleq g$ such that $hklhd g$ and $hcap k$ is contained in $h_{g}$, the core of $h$ in $g$. we characterize the supersolubility of finite groups $g$ with that every maximal subgroup of the sylow subgroups is $nc$-supplemented in $g$.

suppose that $h$ is a subgroup of $g$‎, ‎then $h$ is said to be‎ ‎$s$-permutable in $g$‎, ‎if $h$ permutes with every sylow subgroup of‎ ‎$g$‎. ‎if $hp=ph$ hold for every sylow subgroup $p$ of $g$ with $(|p|‎, ‎|h|)=1$)‎, ‎then $h$ is called an $s$-semipermutable subgroup of $g$‎. ‎in this paper‎, ‎we say that $h$ is partially $s$-embedded in $g$ if‎ ‎$g$ has a normal subgroup $t$ such that $ht...