Exercises - Algebra I Autumn 2014 Exercise 1. Let H be the subgroup of S6 generated by (123), (456), (15)(24)(36). Show that the order of H is 18. Exercise 2. Let G be a finite abelian group, written multiplicatively, and let x be the product of all the elements of G. (1) Show that x2 = 1. (2) Show that if G has a unique element y of order 2, then x = y. (3) Show that if G has more than one element of order 2, then x = 1. Exercise 3. Consider groups of order 2015. (1) Determine all abelian groups among them. (2) Can a group of order 2015 be simple? (3) Can a group of order 2015 be non-abelian? Exercise 4. (1) Consider the cyclic group C2n of order 2n. How many elements does Aut(C2n ) have? (2) Let G be a finite group. Assume that G has a cyclic 2-Sylow subgroup H. Show that the centralizer subgroup ZG (H) of H in G coincides with the normalizer subgroup NG (H) of H in G. Exercise 5. Let G be a finite group with an automorphism σ : G → G which is an involution (i.e., σ ◦ σ = id) and whose only fixed point is the identity element. Prove that G is abelian and it has odd order. Exercise 6. Let T4n := ha, b; a2n = 1, an = b2 , b−1 ab = a−1 i. Show that |T4n | = 4n. Find all irrreducible representations of T4n . Exercise 7. Describe the character table for the groups Z/2Z × Z/2Z, Q8 (the quaternionic groups with 8 elements), A5 , and another group at your choice... Exercise 8. Let G be a finite group and let ρ : G → GL(2, C) be a representation of G. Suppose that there are elements g, h ∈ G such that the matrices ρ(g) and ρ(h) do not commute. Prove that ρ is irreducible. Exercise 9. Suppose that G is the infinite group 1 0 G := : n∈Z , n 1 and let V be the C[G]-module C2 , with the natural multiplication by elements of G. Show that V is not completely reducible. Exercise 10. Give an example of two groups having the same character table but not isomorphic. Exercise 11. Let G be a finite group with conjugacy class representatives g1 , . . . , gk and character table C. Cosnider C as a matrix r × r. Show that det(C) is either real or purely imaginary, and that | det(C)|2 = k Y |CG (gi )|. i=1 Apply this to G = Z/3Z. Exercise 12. Let R be a ring, and let C = R-Mod be the abelian category of left R-modules. Show that the following functors are exact: 1 2 (1) F : C → C, F (N ) = M ⊕ N , for M ∈ C. (2) F : R-Mod → RP -Mod, F (M ) = MP , for P ⊂ R a prime ideal. Exercise 13. Let k be a field. Show that k[x, y]/(x) is not flat as k[x, y]-module. Exercise 14. Let R be a PID. Let I be a prime ideal in the polynomial ring R[x] with the property that I ∩ R 6= 0. Prove that there is an irreducible element π ∈ I ∩ R such that either I = (π) or I = (π, f (x)), for some polynomial f (x) ∈ R[x] which is irreducible mod π, i..e, in the quotient polynomial ring S[x], where S = R/(π). Exercise 15. Let R be a PID, and let b 6= 0 be an element of R. Let F be the field of fractions of R, and let Rb be the subring of F defined by o na Rb := k : a ∈ R and k ≥ 0 . b Show that Rb is a PID. Exercise 16. Let R be a ring with at least two elements. Suppose that, for each nonzero a ∈ R, there exists a unique b ∈ R (which depends on a) such that aba = a. Show that (1) R has no zero divisors. (2) R has a multiplicative identity. (3) Every nonzero element in R has a multiplicative inverse. Exercise 17. Let R = Q[x, y] and let I = (x2 + 1, y − 3) be the ideal generated by the polynomials x2 + 1 and y − 3. Determine whether I is principal, whether I is prime, and whether I is maximal. √ Exercise 18. Let R = Z[ −3]. Let √ I := a + b −3 : a + b is even ⊂ R. Show that I is an ideal in R and determine whether I is a principal ideal or not. Exercise 19. Let S ⊂ N2 be the subset S := (i, j) ∈ N2 : j > 0 ∪ {(0, 0)}. Consider the subset R ⊂ C[x, y] given by X R := ai,j xi y j : ai,j ∈ C . (i,j)∈S (1) Show that R is a commutative C-algebra. (2) Show that R is not finitely generated as C-algebra. Exercise 20. Let R := C[x, y]/(y 2 − x(x − 1)(x + 1)). Show that R is an integral domain, it is not a UFD, but all its localizations at maximal ideals are PID.
© Copyright 2024