Algebra Prelim August 2007
1. Let Gbe a finite group acting transitively on the finite set A. Let Nbe a normal subgroup
of Gand write O1,O2,...,Orfor the distinct N-orbits in A.
(a) For any g∈Gand N-orbit Oi, show gOi:= {ga :a∈ Oi}is an N-orbit and
the action of Gon the set of N-orbits {O1,O2,...,Or}, given by g· Oi=gOi, is
transitive.
(b) For a∈A, let Ga={g∈G|ga =a}be the stabilizer of ain G. Prove r= [G:NGa].
2. (a) Define a solvable group and use your definition to show every dihedral group is
solvable.
(b) Show that if NEGand both Nand G/N are solvable, then Gis solvable.
3. Let R[G] be the group ring of the finite group Gover a commutative ring R. Let
K1,K2,...,Ktbe the different conjugacy classes of G. For each Ki, let Ci=Pk∈Kikbe
the sum of the members of Ki, as an element of R[G]. Prove that the center of R[G] is
the set of sums a1C1+a2C2+· ·· +atCtfor a1, a2, . . . , at∈R.
4. (a) Prove that Z[√2] is a Euclidean domain.
(b) Prove that Z[√−5] is not a UFD by giving an explicit example of nonunique factor-
ization and justifying your example.
5. (a) Give an example of a nonprincipal ideal in Z[x]. Be sure to justify your answer.
(b) Give an example of a nonzero prime ideal in Z[x] which is not a maximal ideal. Be
sure to justify your answer.
6. Let Rbe a commutative ring and Mbe an R-module. If an R-module homomorphism
f:M→Msatisfies f2=f, show M= (ker f)⊕f(M).
