We prove that Lenstra proposition suggesting existence of many counterexamples to Agrawal conjecture is true in a more general case. At the same time we obtain a strictly ascending chain of subgroups of the group (Zp[X]/(Cr(X))) * and state the modified conjecture that the set {X-1, X+2} generate big enough subgroup of this group.

We show that given a collection $$X=\{f_1$$ , ..., $$f_m\}$$ of pure mapping classes on surface S, there is an explicit constant N, depending only X, such their Nth powers $$\{f_1^N$$ $$f_m^N\}$$ generate the expected right-angled Artin subgroup MCG(S). Moreover, we these subgroups are undistorted, and each element pseudo-Anosov largest possible subsurface.

The distinguishing number D(G) of a graph G is the least integer d such that has vertex labeling with labels preserved only by trivial automorphism. Let ? be group acting on set X. for this action X, denoted D? (X), smallest natural k elements X can labeled so any label-preserving element fixes all x ? In particular, if faithful, then preserving identity. paper, we obtain an upper bound knowing...

Let σ = {σi|i ∈ I } be a partition of the set all primes ℙ and G finite group. A ℋ subgroups is said to complete Hall σ-set if every member ≠1 σi-subgroup for some i contains exactly one such that σi ⌒ π(G) ≠ ∅. group σ-primary it σi-group i. subgroup be: σ-permutable in possesses AH x H xA G; σ-subnormal there chain A0 ≤ A1 … At either Ai − 1 ⊴ or /(Ai 1)Ai 1, …, t;

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 $G$.

be found in all classical books on the subject (for example in [4, 44]). For every x ∈ X the isotropy subgroup (also termed fixer or stabilizer) of x is Gx = {g ∈ G : gx = x}. If H ⊂ G is a subgroup of G, then the space fixed by H in X is X = {x ∈ X : Hx = x} = {x ∈ X : H ⊂ Gx}. If X and Y are G-spaces, then a G-map (i.e. an equivariant map) f : X → Y is a map which commutes with the G-action: ...

We say that a subgroup $H$ is isolated in group $G$ if for every $x\in G$ we have either H$ or $\langle x\rangle\cap H=1$. In this short note, describe the set of subgroups finite abelian group. The technique used based on an interesting connection between and function sum element orders

Let G be a connected, simply connected nilpotent Lie group, identified with real algebraic subgroup of UT(n,R), and let Γ lattice in G, π:G→G∕Γ the quotient map. For semialgebraic X⊆G, more generally definable set an o-minimal structure on field, we consider topological closure π(X) compact nilmanifold G∕Γ. Our theorem describes cl(π(X)) terms finitely many families cosets subgroups G. The unde...

‎Let $tau$ be a subgroup functor and $H$ a $p$-subgroup of a finite group $G$‎. ‎Let $bar{G}=G/H_{G}$ and $bar{H}=H/H_{G}$‎. ‎We say that $H$ is $Phi$-$tau$-quasinormal in $G$ if for some $S$-quasinormal subgroup $bar{T}$ of $bar{G}$ and some $tau$-subgroup $bar{S}$ of $bar{G}$ contained in $bar{H}$‎, ‎$bar{H}bar{T}$ is $S$-quasinormal in $bar{G}$ and $bar{H}capbar{T}leq bar{S}Phi(bar{H})$‎. ‎I...