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.
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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