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2003 Nov

2003 November

  1. Let \(H\) be a finite index subgroup of \(G\). Show that there exists a finite index subgroup \(K\) of \(G\) such that \(K \subseteq H\) and \(K\) is normal in \(G\).

  2. Let \(G\) be a group of order 84. Show that \(G\) is not simple.

  3. Let \(G\) be a group and let \(A\) be an abelian normal subgroup of \(G\). Show that there is a nontrivial homomorphism from \(G/A\) to the automorphism group of \(A\).

  4. State what it means for a group to be solvable, and show that any group of order 280 is solvable.

  5. Prove that a group of order 343 has a nontrivial center.