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Sunday, May 11, 2008

Theorem about normal subgroups

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.

\\<br />\text{If x } \in (N\cap H) \text{ and } h \in H \text{ then } \\<br />hxh^{-1} \in H \text{ since } x \in H \text{ and } H \leq G, \text{ and } \\ <br />hxh^{-1} \in N \text{ since } x \in N \text{ and } N \lhd G, \\<br />\text{this shows that }h(N\cap H)h^{-1} \in (N\cap H) \text{ for all }h \in 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.

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