let $g={rm sl}_2(p^f)$ be a special linear group and $p$ be a sylow‎ ‎$2$-subgroup of $g$‎, ‎where $p$ is a prime and $f$ is a positive‎ ‎integer such that $p^f>3$‎. ‎by $n_g(p)$ we denote the normalizer of‎ ‎$p$ in $g$‎. ‎in this paper‎, ‎we show that $n_g(p)$ is nilpotent (or‎ ‎$2$-nilpotent‎, ‎or supersolvable) if and only if $p^{2f}equiv‎ ‎1,({rm mod},16)$‎.

In this paper, we have constituted 3-step general Fibonacci sequences in a nilpotent group with exponent p (p is a prime number) and nilpotency class 4 and given formulas to find the α term of the sequence.

Let p be a prime number. We give the explicit structure of 2-nilpotent multiplier for each finite 2-generator p-group class two. Moreover, 2-capable groups in that are characterized.

Applications of hypergroups have mainly appeared in special subclasses. One of the important subclasses is the class of polygroups. In this paper, we study the notions of nilpotent and solvable polygroups by using the notion of heart of polygroups. In particular, we give a necessary and sufficient condition between nilpotent (solvable) polygroups and fundamental groups.

In the first part of this paper we prove without using the transfer or characters the equivalence of some conditions, each of which would imply p-nilpotence of a finite group G. The implication of p-nilpotence also can be deduced without the transfer or characters if the group is p-constrained. For p-constrained groups we also prove an equivalent condition so that O ′ (G)P should be p-nilpotent...

Let G be a complex simple Lie group and let g be its Lie algebra. Then G has the adjoint action on g. The orbit Ox of a nilpotent element x ∈ g is called a nilpotent orbit. A nilpotent orbit Ox admits a non-degenerate closed 2-form ω called the Kostant-Kirillov symplectic form. The closure Ōx of Ox then becomes a symplectic singularity. In other words, the 2-form ω extends to a holomorphic 2-fo...

Let G be a complex simple Lie group and let g be its Lie algebra. Then G has the adjoint action on g. The orbit Ox of a nilpotent element x ∈ g is called a nilpotent orbit. A nilpotent orbit Ox admits a non-degenerate closed 2-form ω called the Kostant-Kirillov symplectic form. The closure Ōx of Ox then becomes a symplectic singularity. In other words, the 2-form ω extends to a holomorphic 2-fo...

The aim of this paper is to prove Theorem 1.1 below, a generalization to virtually nilpotent spaces of a result of Wilkerson [13] and Sullivan [12]. We recall that a CW complex Y is virtually nilpotent if (i) Y is connected, (ii) 7r~ Y is virtually nilpotent (i.e. has a nilpotent subgroup of finite index) and (iii) for every integer n > 1, zr~Y has a subgroup of finite index which acts nilpoten...

Let $G$ be a reductive group over an algebraically closed field $k$ of separably good characteristic $p>0$ for . Under these assumptions, Springer isomorphism $\phi : \mathcal {N}_{\mathrm {red}}(\mathfrak {g}) \rightarrow {V}_{\mathrm {red}}(G)$ from the nilpotent scheme $\mathfrak {g}$ to unipotent always exists and allows one integrate any $p$ -nilpotent element into One should wonder whe...