1
PROBLEM SET 1 DUE: FEB.24 Problem 1
(1) Let G be the set of integers which can be written as the sum of two squares, that is, G = {x ∈ Z|x = a2+ b2, a, b ∈ Z} . Show that G is a monoid under the multiplication of integers.
(2) Let H be the set of integers which can be written as the sum of four squares, that is, H = {x ∈ Z|x = a2+ b2+ c2+ d2, a, b, c, d ∈ Z} . Show that H is also a monoid under the multiplication of integers.
*(3) Let N be the set of integers which can be written as the sum of eight squares, that is, N = {x ∈ Z|x = a21+ a22+ a23+ a42+ a25+ a26+ a27+ a28, a1, ...., a8∈ Z} . Is it true that N is also a monoid under the multipli- cation of integers?
Problem 2
Consider the lattice Z2⊂ R2, the points on the plane which has inte- gral coordinates, and let C be the convex cone formed by two rays starting from the origin (0, 0) (where we assume these two rays do not lie in the same straight line). Then show that S = C ∩ Z2 is a monoid. Try to prove that the group generated by the monoid S is Z2.
Problem 3
Show that every group of order ≤ 5 is abelian.
Problem 4
Show that there are only two non-isomorphic groups of order 4, namely the cyclic one, and the product of two cyclic groups of order 2.
Problem 5
Let G be a group such that Aut(G) is cyclic. Prove that G is abelian.
Problem 6
Prove the following statement:
2
Let G be a group, S a set of generators for G, and G0 another group.
Let f : S → G0 be a map. If there exists a homomorphism f of G into G0 whose restriction to S is f , then there is only one.
Problem 7
Let G be a group and let H ,H0 be subgroups. By a double coset of H ,H0 one means a subset of G of the form HxH0.
(1) Show that G is a disjoint union of double cosets.
(2) Let {c} be a family of representatives for the double cosets. For each a ∈ G denote by [a]H0 the conjugate aH0a−1 of H0 . For each c we have a decomposition into ordinary cosets
H =[
c
xc(H ∩ [c]H0)
where xc is a family of elements of H ,depending on C. Show that the elements xcc form a family of left coset representatives for H0 in G; that is,
G =[
c
[
xc
xccH0,
and the union is disjoint.
Problem 8
Let G be a group and H a subgroup of finite index. Prove that there is only a finite number of right cosets of H , and that the number of right cosets is equal to the number of the left cosets.
