Normal subgroup of a product of simple groups

  • Thread starter john_nj
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  • #1
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Homework Statement



This is an exercise from Jacobson Algebra I, which has me stumped.
Let G = G1 x G2 be a group, where G1 and G2 are simple groups.
Prove that every proper normal subgroup K of G is isomorphic to G1 or G2.

Homework Equations





The Attempt at a Solution


Certainly the intersection of K with G1 x {1} is normal, and so is isomorphic to the trivial group {1} or to G1. Similarly, the intersection of K with {1} x G2 is isomorphic to {1} or G2. But anyway this falls well short of a solution.

Thanks,

John

Homework Statement





Homework Equations





The Attempt at a Solution


Homework Statement





Homework Equations





The Attempt at a Solution

 

Answers and Replies

  • #2
Office_Shredder
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Well, you've basically got it. Try each of the three cases.... if both intersections are trivial, then what is K? If both intersections are the full group, then what is K? And if only one intersection is trivial, what is K?
 
  • #3
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Thank you. It's now clear.
 

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