The converse of “nilpotent elements are zero-divisors”
For commutative rings $ A$ with identity $ 1\ne0$ , nilpotent elements are zero-divisors. The converse is false, i.e. there is a commutative ring $ A$ with identity $ 1\ne0$ and a zero-divisor $ x$ in...
View ArticleExamples of nilpotent self-distributive algebras
Suppose that $ (X,*,1)$ is an algebra that satisfies the identities $ x*(y*z)=(x*y)*(x*z),x*1=1,1*x=x$ . Define the right powers by letting $ x^{[1]}=x$ and $ x^{[n+1]}=x*x^{[n]}$ . We say that $...
View ArticleIf nilpotent matrix $A$ and $AB−BA$ commute,show that $AB$ is nilpotent.
Let $ A$ and $ B$ be $ n×n$ complex matrices. If $ A$ is an nilpotent matrix, and $ A$ commute with $ AB−BA$ , show that $ AB$ is nilpotent. Equivalently, the question can be expressed as following...
View ArticleCharacterisation of even nilpotent elements in $\mathfrak{sl}_n$
Is there a ”nice” classification of even nilpotent elements in $ \mathfrak{sl}_n,$ using the correspondence between nilpotent elements and partitions of n? By an even element, I mean an element $ e$ ,...
View ArticleGrowth rate of free nilpotent group of rank $r$ and nilpotency index $s$.
Let $ F^{(r)}$ denote the free group of rank $ r$ with generators $ x_1, \dots, x_r$ . Recall a group $ G$ is nilpotent of index $ s$ if $ G_{s+1} = \{e\} $ and $ G_s \neq \{e\}$ (where $ G_i$ denotes...
View ArticleGrowth rate of free nilpotent group of rank $r$ and nilpotency index $s$.
Let $ F^{(r)}$ denote the free group of rank $ r$ with generators $ x_1, \dots, x_r$ . Recall a group $ G$ is nilpotent of index $ s$ if $ G_{s+1} = \{e\} $ and $ G_s \neq \{e\}$ (where $ G_i$ denotes...
View ArticleA nilpotent primitive group is a cyclic group of prime order
There is an exercise in Peter Cameron’s Permutation Groups that a nilpotent primitive group is cyclic of prime order. However I can only prove the finite case, by writing the group as a direct product...
View Articlesummarizing the kernel and nilpotent element of 0
If the kernel only contains one nilpotent of 0, does that make the kernel trivial? I was just wondering if this holds true. The post summarizing the kernel and nilpotent element of 0 appeared first on...
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