MATH 422 Midterm: MATH 422+501 2012 Winter Test 1

• School: UBC
• Department: Mathematics
• Course: MATH 422

Document Summary

Exam rules: you can refer to any result that was proved in class or that appeared in a homework. Ask if you want to refer to some other result: there are 16 problems in this exam. Let g be a nite abelian group that is not cyclic. Prove that g contains a subgroup isomorphic to ip ip for some prime p. Let p be a normal sylow p-subgroups of a nite group g. prove that if : g g is a group homomorphism, then (p ) p . Prove that a group of order 31 32 cannot be simple. Let g be a nite group, such that the group of automorphisms aut(g) is cyclic. Prove that g is abelian. (hint: there is a homomorphism g aut(g). Study the kernel and the image of this homomorphism. ) Prove that there is no simple group g of order 300. (hint: let g act on its sylow 5-subgroups. )