We describe a new order-theoretic property of the commutator for finite algebras. As a corollary we show that any right nilpotent congruence on a finite algebra is left nilpotent. The result is false for infinite algebras and the converse is false even for finite algebras. We show further that any solvable E-minimal algebra is left nilpotent, any finite algebra whose congruence lattice contains...

In this paper we study degenerations of nilpotent Lie algebras. If λ, μ are two points in the variety of nilpotent Lie algebras, then λ is said to degenerate to μ , λ→deg μ , if μ lies in the Zariski closure of the orbit of λ . It is known that all degenerations of nilpotent Lie algebras of dimension n < 7 can be realized via a one-parameter subgroup. We construct degenerations between characte...

the zero-divisor graph of a commutative ring r with respect to nilpotent elements is a simple undirected graph $gamma_n^*(r)$ with vertex set z_n(r)*, and two vertices x and y are adjacent if and only if xy is nilpotent and xy is nonzero, where z_n(r)={x in r: xy is nilpotent, for some y in r^*}. in this paper, we investigate the basic properties of $gamma_n^*(r)$. we discuss when it will be eu...

Network architecture can lead to robust synchrony in coupled systems and, surprisingly, to codimension one bifurcations from synchronous equilibria at which the associated Jacobian is nilpotent. We prove three theorems concerning nilpotent Hopf bifurcations from synchronous equilibria to periodic solutions, where the critical eigenvalues have algebraic multiplicity two and geometric multiplicit...

I would like to discuss two results about finite groups: (0) All finite groups of odd order are solvable. (F) A finite group is nilpotent if it admits a fixed point free automorphism of prime order. Walter Feit and I proved (0) after a prolonged joint effort [5]. A critical special case of (0) was proved by Walter Feit, Marshall Hall, Jr. and me [4]. A slightly stronger result than (F) is in th...

We associate a graph NG with a group G (called the non-nilpotent graph of G) as follows: take G as the vertex set and two vertices are adjacent if they generate a non-nilpotent subgroup. In this paper we study the graph theoretical properties of NG and its induced subgraph on G\nil(G), where nil(G) = {x ∈ G | 〈x, y〉 is nilpotent for all y ∈ G}. For any finite group G, we prove that NG has eithe...

Fix a prime p, and let A be the polynomial part of the dual Steenrod algebra. The Frobenius map on A induces the Steenrod operation P̃0 on cohomology, and in this paper, we investigate this operation. We point out that if p = 2, then for any element in the cohomology of A, if one applies P̃0 enough times, the resulting element is nilpotent. We conjecture that the same is true at odd primes, and t...