Problem 1. Prove that in a group (a1 )1 = a.
Proof. Let G be a group. Suppose a G, then there exists b G such that ab = e. The
element b is denoted by a1 . So, a1 = b. By taking the inverse of both sides, it follows
Problem 1. For every positive integer n, prove that
1 + 2 + . + n =
n(n + 1)
Proof. We will proceed using Mathematical Induction. For the base case, when n = 1
notice that 1 = 1(1+1)
. Now for the inductive step, we w
Problem 1. Give two reasons why the set of odd integers under addition is not a group.
The set of odd integers does not form a group under addition because it is not closed
under the operation of addition. That is, the su
Problem 1. Determine the subgroup lattice for Z8 .
The set of all divisors of 8 is cfw_1, 2, 4, 8. By the corollary, Subgroups of Zn , we obtain
the following subgroups of Z8 :
h8i = cfw_0
h4i = cfw_0, 4
h2i = cfw_0, 2, 4