Durian & Group
Those durians which have no kernel (臼籽 \jiu-zi) but meat are excellent – they are SIMPLE.
Eating one kernel (Normal Subgroup) is enough to know whether the Durian (Group) is D24, D18 or 猫山王 (Mountain Cat King) or 黑刺 (Black Thorns) species.
Knowing the kernel 核of a fruit will allow biologists to understand the whole fruit.
In Group, a kernel of group homomorphism is a Normal subgroup, hence will let us know the whole group.
Normal subgroup is the important essence revealing the whole group.
First, you must realize what a Group is? Group is a set with an operation (Transformation) acting on its elements such that
Within a Group G, it has some subgroups other than two trivial subgroups (Identity and itself G). These subgroups obey CANI within itself like G.
Among the subgroups, there is one type called “invariant” (Galois’s discovered it Invariant, or Conjugate subgroup, Americans call it Normal subgroup), which keeps the “symmetry” (meaning, no change after transformation, eg. rotate a circle around its center, the circle is not changed by the Rotation).
The Normal subgroups N reveal the “symmetry” of the Group – that is what Group is for studying Symmetry, so the importance of knowing N.
With the Normal Subgroup found, we can do the followings:
2) When compare two Groups G and G’ by a mapping f (called Group Homomorphism, ie see if they are ‘similar’),
Group Theorem tells us the Kernel of G (Ker f) is a Normal Subgroup of G
ie ∀x ∈ Kerf, f(x) = e’
3) Normal subgroup can have a smaller Normal subgroup, which in turn has its smaller Normal Subgroup, and so on.
When there is no more Normal Subgroup (Simple Group), then the Group is not Solvable (or Soluable) ie no Solution. This was Galois proof of Quintic Equations have no radical Solution because Alternating Group A5 is Simple Group.