Let $x=uv, u\in S$, $v\in \overline S$ and $y=wz, w\in S$, $z\in \overline S$. Removal of $x,y$ make the graph disconnected. Suppose there exists a cycle $\mathscr C$ consists of $x$ without $y$. So, there exists a $u-v$ path other than edge $x$. Let it be $u e'_1u_1 u_2e'_2 ...u_k e'_k v$. Ther...

One can give the cross product as a Lie bracket on $\mathbb{R}^3$ and the matrix commutator to $\mathbb{R}^{n^2}$ ($n \ge 2$). They both give a perfect Lie algebra structure. However, every Lie algebra of dimension $1$ is abelian, and for $2$-dim we can write any nontrivial Lie bracket as $[x, y...

Lie algebra $\mathfrak{g}$ is perfect if and only if $\mathfrak{g}=[\mathfrak{g},\mathfrak{g}]$. And we know semisimple Lie algebra must satisfy the $\mathfrak{g}=[\mathfrak{g},\mathfrak{g}]$ then it's perfect. What is the example of perfect Lie algebra but not semisimple? And what's the suffi...

If L is a semisimple lie algebra then L=[L,L]. Is the opposite true?

Possible Duplicate: Why isn't reflexivity redundant in the definition of equivalence relation? We had a heated discussion in class today and i still cant be sure if the professor was any good with the solution. The question is: If a relation is symmetric and transitive, then it wil...