Exercises - Algebra I Autumn 2014

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.