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2006 Apr

2006 April

Instructions. Do as many problems as you can. You are not expected to do all of the problems. You may use earlier parts of a problem to solve later parts, even if you cannot solve the earlier part; however, complete solutions are preferred. Most importantly, give careful solutions.

  1. Show that there is no simple group of order 992.

  2. Let \(G\) be a nonabelian simple group. Let \(S_n\) be the symmetric group of all permutations on an \(n\)-element set, and let \(A_n\) be the alternating group.\ a. Show that if \(G\) is a subgroup of \(S_n\), \(n\) finite, then \(G\) is a subgroup of \(A_n\).\ b. Let \(H\) be a proper subgroup of \(G\), and, for \(g\in G\), let \(\lambda_g\) be the map of the set of left cosets of \(H\) onto themselves defined by \(\lambda_g(xH) = gxH\). Show that the map \(g\mapsto \lambda_g\) is a monomorphism (injective homomorphism) of \(G\) into the group of permutations of the set of left cosets of \(H\).\ c. Let \(H\) be a subgroup of \(G\) of finite index \(n\) and assume \(n>1\) (so \(H \neq G\)). Show that \(G\) can be embedded in \(A_n\).\ d. If \(G\) is infinite, it has no proper subgroup of finite index.\ e. There is no simple group of order 112.

  3. a. Let \(\alpha\) be an element of the symmetric group \(S_n\) and let \((i_1 i_2 \dots i_k)\) be a cycle in \(S_n\). Prove that \(\alpha^{-1}(i_1 i_2 \dots i_k)\alpha= (i_1\alpha i_2 \alpha \dots i_k\alpha)\). (Note that it is assumed that permutations act on the right, so \(\alpha\) maps \(i\) to \(i\alpha\).)\ b. Show that \(A_4\) is not simple.\ c. Show that any five-cycle \(\sigma \in S_5\) and any two-cycle \(\tau \in S_5\) together generate \(S_5\).

  4. If \(A\) and \(B\) are subgroups of a group \(G\), let \(A\vee B\) be the smallest subgroup containing both, and let \(AB = \{ab : a\in A, b\in B\}\).\ a. Show that if \(A\) is a normal subgroup then \(AB = A\vee B\) and that, if both \(A\) and \(B\) are normal, then \(A\vee B\) is normal.\ b. If \(A, B\) and \(C\) are normal subgroups of \(G\) and \(C\subseteq A\), prove Dedekind's modular law: \(A \wedge (B\vee C) = (A\wedge B) \vee C\).

  5. Let \(G\) be a group and let \(Z\) be its center.\ a. Show that if \(G/Z\) is cyclic then \(G\) is abelian.\ b. Show that any group of order \(p^2\), where \(p\) is a prime, is abelian.\ c. Give an example of a non-abelian group \(G\) where \(G/Z\) is abelian.\ d. Let \(\varphi\) be a homomorphism from \(G\) onto \(K\), where \(K\) is an abelian group. Let \(N\) be the kernel of \(\varphi\) and suppose \(N\) is contained in \(Z\). Suppose that there is an abelian subgroup \(H\) of \(G\) such that \(\varphi(H) = K\). Show \(G\) is abelian.

  6. Let \(G\) be a finite group and let \(H \triangleleft G\) be a normal subgroup. Show that \(G\) is solvable if and only if \(H\) and \(G/H\) are solvable.