PMATH 347 Groups and Rings, Assignment 1
Due Fri May 24
1: (a) Find the number of binary operations on Zn .
(b) Find the number of commutative binary operations on Zn .
(c) Find the number of binary operations on Zn with the left-cancellation property, th
PMATH 347 Groups and Rings, Assignment 5
Due Mon July 29
1: (a) Find all the units and all the zero divisors in the ring Z4 [i].
(b) Without proof, list all of the subrings of Z4 [i].
(c) Without proof, list all of the ideals in Z4 [i].
(d) Find all prime
PMATH 347 Groups and Rings, Assignment 4
Due Fri July 19
1: (a) Find a group of the form Zn1 Znl , with ni ni+1 for all i, which is isomorphic to Z18 Z60 Z70 Z100 .
(b) Find the number of distinct abelian groups of order 2,000,000 (up to isomorphism).
(c)
PMATH 347 Groups and Rings, Assignment 3
Due Fri June 28
1: (a) Find Hom(Zn , Zm ) .
(b) Find the number of subgroups K Zn such that K = Ker() for some Hom(Zn , Zm ).
(c) Show that Aut(Zn ) = Un .
(d) Find Hom(Zn Zm , Zl ) .
2: (a) Show that for n 3 we ha
PMATH 347 Groups and Rings, Assignment 2
1: The centre of a group G is the set Z (G) = a G ax = xa for all x G .
(a) Show that Z (G) G.
(b) Find Z (Dn ).
(d) Find Z (Sn ).
(c) Find Z GLn (R) .
2: (a) Find the number of cyclic subgroups of Z9 Z15 .
(b) Fin
PMATH 347 Groups and Rings, Assignment 6
Not to be handed in
1: Let R be a unique factorization domain. Prove each of the following statements.
(a) Every irreducible element in R is prime.
(b) For all a1 , a2 , a3 , R with a1 a2 a3 , there exists n 1 with
PMATH 347 Groups and Rings, Solutions to Assignment 6
1: Let R be a unique factorization domain. Prove each of the following statements.
(a) Every irreducible element in R is prime.
Solution: Let a R be irreducible. Let x, y R. Suppose a xy , say xy = ab
PMATH 347 Groups and Rings, Solutions to Assignment 5
1: (a) Find all the units and all the zero divisors in the ring Z4 [i].
Solution: Let a, b Z4 with a + ib = 0 Z4 [i]. When a = b = 0 mod 2 we have (a + ib)(a ib) = a2 + b2 =
0 + 0 = 0, so a + ib is a z
PMATH 347 Groups and Rings, Solutions to Assignment 4
1: (a) Find a group of the form Zn1 Znl , with ni ni+1 for all i, which is isomorphic to Z18 Z60 Z70 Z100 .
Solution: Z18 Z60 Z70 Z100 = (Z2 Z9 ) (Z4 Z3 Z5 ) (Z2 Z5 Z7 ) (Z4 Z5 )
(Z2 Z2 Z4 Z4 ) (Z3 Z9
PMATH 347 Groups and Rings, Solutions to Assignment 2
1: The centre of a group G is the set Z (G) = a G ax = xa for all x G .
(a) Show that Z (G) G.
Solution: We have e Z (G) because ex = x = xe for all x G. If a G and b G so that ax = xa and
bx = xb for
PMATH 347 Groups and Rings, Solutions to Assignment 3
1: (a) Find Hom(Zn , Zm ) .
Solution: Recall that the group homomorphisms : Zn Zm are the maps a where a Zm with na = 0.
Let us determine which elements a Zm satisfy na = 0. Let d = gcd(n, m) and say n