Let be a Lie algebra and its Killing form. Is the kernel of always equal to the radical of ?
Let be a connected Lie subgroup of . Then commutes with all of iff it commutes with all of .
Is there a compact 3-manifold having as boundary?
A plane flies 100 km north, 100 km east, 100 km south and then 100 km west. Give a necessary and sufficient condition for it to land on the same spot where it took off.
Prove the following two statements and conclude that the fundamental group of a (connected) topological group is abelian:
1) A discrete normal subgroup of a connected topological group is central.
2) If is the universal covering, the total space admits a topological group structure such that is a group morphism. The kernel of is then isomorphic to the fundamental group of .
The exponential map is surjective for but not for .
If one believes that is a vector space over the field with one element , then a permutation of is just a complete flag. Concretely, a permutation corresponds to the chain of subspaces .
As we all know, there are permutations of , or equivalently complete flags of . Give a formula for the number of complete flags of , which will be a -analogue for the factorial.