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## Sunday, May 11, 2008

Normal subgroups are very important objects in Group Theory. One of the 'must-never-forget'-theorems is the following.

Let N be a normal subgroup of a group G and H be any subgroup of G. Then the intersection of H and N is a normal subgroup of H.

For the proof we use the following theorem.
A subgroup H is a normal subgroup in G if gH=Hg for all elements g in G.

$\\&space;&space;\text{If&space;x&space;}&space;\in&space;(N\cap&space;H)&space;\text{&space;and&space;}&space;h&space;\in&space;H&space;\text{&space;then&space;}&space;\\&space;&space;hxh^{-1}&space;\in&space;H&space;\text{&space;since&space;}&space;x&space;\in&space;H&space;\text{&space;and&space;}&space;H&space;\leq&space;G,&space;\text{&space;and&space;}&space;\\&space;&space;&space;hxh^{-1}&space;\in&space;N&space;\text{&space;since&space;}&space;x&space;\in&space;N&space;\text{&space;and&space;}&space;N&space;\lhd&space;G,&space;\\&space;&space;\text{this&space;shows&space;that&space;}h(N\cap&space;H)h^{-1}&space;\in&space;(N\cap&space;H)&space;\text{&space;for&space;all&space;}h&space;\in&space;H.\\$

In easy to remember math: "The intersection of a normal subgroup with another subgroup is normal in that subgroup." ( H&N is N(ormal)in H )

If this seems difficult: this theorem becomes trivial real fast.

## Welcome to The Bridge

Mathematics: is it the fabric of MEST?
This is my voyage
My continuous mission
To uncover hidden structures
To create new theorems and proofs
To boldly go where no man has gone before

(Raumpatrouille – Die phantastischen Abenteuer des Raumschiffes Orion, colloquially aka Raumpatrouille Orion was the first German science fiction television series. Its seven episodes were broadcast by ARD beginning September 17, 1966. The series has since acquired cult status in Germany. Broadcast six years before Star Trek first aired in West Germany (in 1972), it became a huge success.)