Do all problems

- Prove that a group of order
2256 = 47*48 cannot be simple.
- Let
*G*= <*x*,*y*|*x*^{7}=*y*^{3}= 1,*yxy*^{-1}=*x*^{2}>.- (i)
- Prove that every element of
*G*can be written in the form*x*^{i}*y*^{j}where*i*,*j*are non-negative integers. - (ii)
- Prove that
*G*has order at most 21. - (iii)
- Prove that there is a homomorphism
q :
*G*- >*S*_{7}such thatq*x*= (1 2 3 4 5 6 7), q*y*= (2 3 5)(4 7 6). - (iv)
- Prove that
*G*has order 21.

- Let
*R*be a UFD. Suppose that for every coprime*p*,*q**R*, the ideal*pR*+*qR*is principal. Prove that for every*a*,*b**R*, the ideal*aR*+*bR*is principal. (Coprime means that the greatest common divisor of*p*,*q*is 1.) - Let
*R*be the ring**Z**/4**Z**and let*M*be the ideal 2**Z**/4**Z**. Prove that*M**M*@*M*as*R*-modules. - Let
*R*be a PID and let*M*,*N*be*R*-modules. Suppose*M*is finitely generated and*M**M*@*N**N*. Prove that*M*@*N*. - Let
*K*be a field of characteristic zero, let*f**K*[*x*] be an irreducible polynomial, let*L*be a splitting field for*f*over*K*, and let a_{1},a_{2},a_{3},a_{4}*L*be the roots of*f*. Suppose [*L*:*K*] = 24 (i.e. dim_{K}*L*= 24).- (i)
- If
1
__<__*i*,*j*__<__4 are integers, prove that*L*=/=*K*(a_{i},a_{j}). - (ii)
- Prove that
*L*=*K*[a_{1}+ 2a_{2}+ 3a_{3}].

- Let
*k*be an algebraically closed field, let*n*be a positive integer, and let*U*,*V*be affine algebraic sets in*k*^{n}(so*U*is the zero set of a collection of polynomials in*k*[*x*_{1},...,*x*_{n}]). Suppose*U*/\*V*= f. Prove that I(*U*) + I(*V*) =*k*[*x*_{1},...,*x*_{n}] (where I(*U*) is the set of all polynomials in*k*[*x*_{1},...,*x*_{n}] which vanish on*U*).