Why Read This Book? - Index of

EXERCISE 8.2.23 If G is a group and a ∈ G, then (a) is abelian.
8.2 Subgroups 259
EXERCISE 8.2.24 Suppose G is an abelian group, and a, b ∈ G. Then
({a, b}) ={a m b n : m, n ∈ Z} (8.30)
EXERCISE 8.2.25 Write Eq. (8.30) in its additive form.
EXERCISE 8.2.26 Consider the group (Z, +, 0, −), and let a and b be nonzero
integers and g = gcd(a, b). Then ({a, b}) = (g).
EXERCISE 8.2.27 Suppose G is a group and a ∈ G. Let m and n be positive
integers, and suppose gcd(m, n) = g. Then ({a m ,a n }) = (a g ).
EXERCISE 8.2.28 For the additive group of integers, determine the following
without proof.
(a) (4) ∩ (6)
(b) (10) ∩ (3)
(c) (8) ∩ (16)
(d) (12) ∩ (16) ∩ (28)
(e) ({4, 6})
(f) ({4, 7})
(g) ({12, 16, 28})
If G is a group and there exists some element g such that (g) = G, then G is
said to be a cyclic group and g is called a generator of the group. There are some
nifty little theorems about cyclic groups. For example, by Exercise 8.2.23, a cyclic
group is abelian. Here is another.
EXERCISE 8.2.29 A cyclic group is countable.
EXERCISE 8.2.30 Determine with explanation whether each of the following
groups is cyclic.
(a) (Z, +, 0, −)
(b) The group from Example 8.1.9
(c) (R × , ×, 1, −1 )
(d) The quaternion group from Example 8.1.18
(e) (Q, +, 0, −)