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2008 Apr

2008 April

  1. Prove that \(S_4\) is solvable but not nilpotent.

  2. List all abelian groups of order 100. Prove that your list is complete.

  3. How man elements of order 7 are there in a group of order 168?

  4. Let \(G' < G\) be the commutator subgroup of a finite group \(G\). Let \(Z\) be the center of \(G\). Let \(p\) be a prime number. Suppose that \(p\) divides \(|Z|\) but does not divide \(|Z\cap G'|\). Show that \(G\) has a subgroup of index \(p\).

  5. Assume \(G\) is a finite group, \(p\) the smallest prime dividing the order of \(G\), and \(H\) a subgroup of index \(p\) in \(G\). Prove that \(H\) is a normal subgroup of \(G\).