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Exercise Batch 7 Taken from John B. Fraleigh, A First Course in Abstract Algebra, Fifth Edition Exercise Set 1 From Page 170 ff: 1-15, 17-20, 22, 23, 24-33, 35, 36, 40-44. As we said in class, if G is a group, and if X is a set, then G acts on X if there is a homomorphism G → SX . The kernel of this homomorphism is usually called the kernel of the action. If the kernel is trivial, then we say that the action is faithful. In each case below, I’ve sketched a group action. You should state carefully what the set X is, prove that what is described is really a group action, and compute the kernel of the action. Also try to determine whether the action is transitive. (In many cases, there won’t be enough information to tell whether the action is transitive.) 1. G acts on itself by left multiplication. 2. G acts on itself by conjugation, i.e., if g, x ∈ G, then g : x 7→ gxg −1 . 3. Let H ≤ G; G acts on left cosets of H by left multiplication. 4. G acts on the set of its subgroups by conjugation. 5. Let G ≤ Sn , and fix an integer k, 0 ≤ k ≤ n. Then G acts on the nk subsets of Nn of size k. If G = Sn is the action transitive? What about if G = An ? 6. Let G ≤ GLn (F), and fix an integer k, 0 ≤ k ≤ n. Then G acts on the k-dimensional subspaces of the vector space of n × 1 column vectors with entries in the field F. If G = GLn (F) is the action transitive? Exercise Set 2 From Page 183 ff: 4-8, 11, 12-14, 18-21, 23.