Answer all questions

- (a)
- Let
*M*be a left*R*-module, and let*A*and*B*be Artinian submodules. Show that*A*+*B*is an Artinian*R*-submodule. - (b)
- If
*R*is also left Noetherian and*M*is finitely generated, show that*M*has a unique maximum Artinian submodule*A*(*M*) and that*A*(*M*/*A*(*M*)) = 0.

- Let
*A*be an abelian group with no elements of infinite order. Suppose that every element of prime order is of order 3. Show that the order of every element is a power of 3. (Hint: do finitely generated abelian groups first.) - Let
*G*be a simple group of order 144.- (a)
- Prove that a group of order 18 has exactly one Sylow
3-subgroup.
- (b)
- If
*H*is a proper subgroup of*G*, show that |*H*|≤26. - (c)
- If
*P*and*Q*are distinct Sylow 3-subgroups of*G*, show that |*P*∩*Q*| = 1. (If |*P*∩*Q*| > 1, consider*N*_{G}(*P*∩*Q*)).

- Prove that a group of order 765 is abelian.
- Let
*f*(*x*) in**Q**[*x*] be an irreducible polynomial of degree 5. Suppose*a*and*b*are distinct roots and that**Q**(*a*) =**Q**(*b*). Show that**Q**(*a*) is a normal extension of**Q**. - Let
*S*⊆**Z**[*x*_{1},*x*_{2},...,*x*_{n}]. Prove that there is a smallest principal ideal containing*S*. If this ideal is generated by α, show that α**Q**[*x*_{1},*x*_{2},...,*x*_{n}] is the smallest principal ideal in**Q**[*x*_{1},*x*_{2},...,*x*_{n}] containing*S*. - Let
*V*be a vector space over*R*, and let*T*:*V*->*V*be a linear transformation. Describe how*V*can be made into an*R*[*x*]-module.Now suppose there are

*v*_{1},...,*v*_{n}in*V*such that {*T*^{i}(*v*_{j}) |*i*= 0, 1,...,*j*= 1, 2,...,*n*} span*V*as a vector space.- (a)
- Prove that
*V*is a finitely generated*R*[*x*]-module. - (b)
- If
*T*is onto, show that*V*cannot have a summand isomorphic to*R*[*x*]. - (c)
- If
*T*is onto, show that*V*is finite dimensional.

- (a)
- Let
*A*= {ι,(1 2)(3 4)} and*V*= {ι,(1 2)(3 4),(1 3)(2 4),(1 4)(2 3)} in*S*_{4}. Show that*A*is normal in*V*and*V*is normal in*S*_{4}, but*A*is not normal in*S*_{4}. - (b)
- Give an example of fields
*F*⊆*K*⊆*L*such that*K*a normal extension of*F*and*L*a normal extension of*K*, but*L*is not a normal extension of*F*.

Peter Linnell 2012-01-15