Intersection of kernels of all irreducible Brauer characters

Question

We now intersection of kernels of all irreducible characters of a group is trivial. Also p-core(G) is contained in kernel of every irreducible Brauer character.

What is the intersection of kernels of all irreducible Brauer characters???

Can we show it is minimal normal subgroup of group?

Answer 1

Here is a brief argument: since the Brauer table, as a matrix, has full rank, every $p’$ element has a nontrivial image under some irreducible representation. That means that the intersection of all kernels can only contain $p$ elements, and thus can’t be larger than $O_p$.