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2004 Nov

2004 November

  1. a. Carefully state (without proof) the three Sylow theorems.\ b. Prove that every group of order \(p^2 q\), where \(p\) and \(q\) are primes with \(p< q\) and \(p\) does not divide \(q-1\), is abelian.\ c. Prove that every group of order 12 has a normal Sylow subgroup.

  2. a. Among finite groups, define nilpotent group, in terms of a particular kind of normal series. State two conditions on \(G\) which are equivalent to the condition that \(G\) is nilpotent.\ b. Prove that the center \(Z(G)\) of a nilpotent group is non-trivial.\ c. Give an example which shows that \(N \triangleleft G\) with both \(N\) and \(G/N\) nilpotent is not sufficient for \(G\) to be nilpotent.

  3. a. If \(K, L\) are normal subgroups of \(G\) prove that \(G/K\cap L\) is isomorphic to a subgroup of \(G/K \times G/L\) (the external direct product). What is the index of this subgroup in \(G/K \times G/L\), in terms of \([G:K], [G:L]\) and \([G:KL]\)?\ b. Prove either direction of: If \(N\triangleleft G\), then \(G\) is solvable if and only if both \(N\) and \(G/N\) are solvable.

  4. a. If \(H\) is a subgroup of \(G\) and \([G:H]=n \geq 2\), prove that there exists a homomorphism \(\rho\) from \(G\) into \(S_n\), the group of all permutations of an \(n\)-element set. Show that the kernel of \(\rho\) is contained in \(H\), and the image of \(\rho\) is a transitive subgroup of \(S_n\).\ b. If \([G:H]=n\) and \(G\) is simple, then \(G\) is isomorphic to a subgroup of \(A_n\), the subgroup of all even permutations.\ c. Every group of order \(2^3 \cdot 3^2 \cdot 11^2\) is solvable.

  5. Let \(G\) be a finite group, and suppose the automorphism group of \(G\), \({{\mathrm{Aut}}}(G)\), acts transitively on \(G\setminus \{1\}\). That is, whenever \(x, y \in G\setminus \{1\}\) there exists an automorphism \(\alpha\in {{\mathrm{Aut}}}(G)\) such that \(\alpha(x) = y\). Prove that \(G\) is an elementary abelian \(p\)-group for some prime \(p\), by proving that:\ a. All non-identity elements of \(G\) have order \(p\), for some prime \(p\).\ b. Here \(Z(G) \neq 1\), and the center of every group is a characteristic subgroup.\ c. Thus \(G\neq Z(G)\), i.e., \(G\) is abelian.

  6. Suppose \(N\) is a normal subgroup of \(G\). \(C_G(N)\) denotes \(\{g\in G \mid g^{-1}ng = n \text{ for each } n\in N\}\).\ a. Prove that \(C_G(N)\triangleleft G\), and if \(C_G(N) = \{1\}\) then \(|G|\) divides \(|N|!\).\ b. If \(N\) is also cyclic, prove that \(G/C_G(N)\) is abelian, and hence that \(G' \leq C_G(N)\), where \(G'\) is the commutator subgroup of \(G\).