MATH 241 Final, Fall 2003

tee emmnpamane ea “ OW edd | LIBRARY | OF auayT Math 241 Final Exam. (Prof. N. Nahlus) Jan. 27, 2004 PartI: (80%) _Answer any 8 of the following 10 problems. 1. Show that Z3 x Zsx Z2s = Z309 by finding a (well-defined) isomorphism between the groups. 2. Let H be a normal subgroup of a finite group G where 0o(G)=170 & o(H) =10 . If x © H, showthat x€H. (Whatis o(G/H) 7) 3. Answer any part (i) Prove the 1* theorem of Sylow (assuming Cauchy’s Theorem for abelian groups). OR (ii) Prove the Orbit-Stabilizer theorem for a finite group G acting on a set S. ( |orbit(s)| = ....) 4, Let A and B are normal subgroups of a group G. Show that AB/A = BI(AMB). . Construct a non-abelian group of order 55. ; (Hint: Use the format G=<a,b} a!!=1, b°=1, and bab” =a! > witha suitable i.) . LetG bea group of order 55 such that Z(G) # {1}. Show that G is abelian. (Hint: G is abelian iff. . Let G be a finite abelian group G of order 100. Show that the mapping f:G— G given by f(x) = x!7 isan isomorphism. 8. Show that every field R has no ideals other than 0 & R. 9. If 0 & Rare the only ideals of a commutative ring R with 1, show that R is a field. 10. Prove any part of the following (i) If Lis a maximal ideal of a commutative ring R with 1, show that R/I is a field. OR (ii) If lis a prime ideal of a commutative ring R with 1, show that R/I is an integral domain. Part II (20%) Obligatory A) Let H bea subgroup of Gof index 7. If 7 is the smallest prime dividing the order of G, show that H is normal in G. (Hint: Consider the morphism f:G — Perm(G/H) givenby f(g)=L,. Show that H=Kerf.) B) Let A and B be two normal subgroups of G. Consider the subgroup F= {(xA, yB): xAB=yAB} within the group G/A x G/B. Show that G/(ANB)=F. C) LettH<A<G whereA isacyclic normal subgroup of a group G. Show that H is normal in G. (Hint: Every subgroup of a cyclic group is also cyclic)