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Given a homomorphism f:S -> S', the kernel consists of all elements of S that are mapped to the identity elemnt of S'.
The virtual kernel is a kernel that can be used in unbuntu guest. It is a very lean kernel, this helps in reducing overhead.
It installs the server kernel via a new name.
The Kernel
Mac OS X is based on the XNU kernel, a microkernel Mach kernel with a BSD userland, which makes Mac OS X's kernel a hybrid-kernel.
Ubuntu uses the Linux kernel, which is a monolithic kernel with loadable modules.