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2002 Nov

2002 November

  1. Let \(A\) be a finite abelian group. Let \(n\) be the order of \(A\) and let \(m\) be the exponent of \(A\) (\(m\) is the least positive integer such that \(ma = 0\) for all \(a\in A\)).\ a. Show that \(m\mid n\) (\(m\) is a factor of \(n\)).\ b. Show that \(A\) is cyclic if and only if \(m=n\).

  2. Let \(G\) be a finite group. Let \(H\leq G\) be a subgroup.\ a. Show that the number of distinct conjugates of \(H\) in \(G\) is \(|G|/|N(H)|\), where \(N(H)\) is the normalizer of \(H\) in \(G\) and \(| \cdot |\) indicates order.\ b. Show that \(G = \bigcup_{g\in G} gH g^{-1}\) if and only if \(H = G\).\ c. Deduce that \(G\) is generated by a complete set of representatives of the conjugacy classes of elements of \(G\).

  3. Let \(p, q\) and \(r\) be distinct prime numbers.\ a. List, up to isomorphism, all abelian groups of order \(p^4\).\ b. Up to isomorphism, how many abelian groups of order \(p^4q^4 r^3\) are there?

  4. Let \(G\) be a nonsolvable group of least order among nonsolvable groups. Show that \(G\) is simple.

  5. Let \(G\) be a group of order \(p^n m\), where \(p\) is a prime number, \(n\geq 1\) and \(p\) is not a factor of \(m\). Let \(r\) be the number of Sylow \(p\)-subgroups of \(G\) and let \(E = \{K_1, \dots, K_r\}\) be the set of Sylow \(p\)-subgroups of \(G\). Let \(N(K_i)\) be the normalizer of \(K_i\) in \(G\), and let \(G\) act on \(E\) by conjugation.\ a. Show that \(\bigcap_{i=1}^r N(K_i)\) is a normal subgroup of \(G\).\ b. Show that \(|\bigcap_{i=1}^r N(K_i)|\geq p^n/(r-1)!\)\ c. Deduce that a group of order 48 is not simple.

  6. Show that a finite group \(G\) is nilpotent if and only if \(G\) is isomorphic to the product of its Sylow \(p\)-subgroups.