The University of Hong Kong
DEPARTMENT OF MATHEMATICS
MATH2301 Algebra I
Suggested solutions for class test I
1. (20 points) Let G be a group such that no element in G has order 2. Prove
the order of G must be an odd number.
Proof. Suppose no element in G
The University of Hong Kong
DEPARTMENT OF MATHEMATICS
MATH2301 Algebra I
Suggested solutions for Assignment II
Let G be a group with a subgroup H.
1. Let a, b G. If Ha = Hb, prove Ha is disjoint with Hb.
Proof. Assume Ha Hb = , and then there exist some h
The University of Hong Kong
DEPARTMENT OF MATHEMATICS
MATH2301 Algebra I
Suggested solutions for Assignment I
1. Determine all subgroups of (Z, +).
Proof. Since Z is cyclic, then all the subgroups are cyclic. Let H1 = n
and H2 = m be a subgroup generated
The University of Hong Kong
DEPARTMENT OF MATHEMATICS
MATH2301 Algebra I
Assignment II
Due Monday at 5 pm, Oct. 03, 2011.
Let G be a group with a subgroup H.
1. Let a, b G. If Ha = Hb, prove Ha is disjoint with Hb.
2. Prove H is a normal subgroup of G.
1)
The University of Hong Kong
DEPARTMENT OF MATHEMATICS
MATH2301 Algebra I
Assignment I
Due Wednesday at 5 pm, Sept. 21, 2011.
1. Determine all subgroups of (Z, +).
2. Dene n = cfw_z C | z n = 1.
1) Prove that n is a group with the usual multiplication as i