Why Read This Book? - Index of

254 Chapter 8 Groups
(a) Explain why zero is not in G.
(b) Show that G is a subgroup of (R × , ×, 1, −1 ) by showing it has properties
H1–H3.
EXERCISE 8.2.8 Define the center of a group G to be the set of all elements
that commute with all elements of G. That is, the center is
Show that the center of G is a subgroup of G.
{a ∈ G : a ∗ x = x ∗ a for all x ∈ G} (8.20)
EXERCISE 8.2.9 Suppose {Hα}α∈A is a family of subgroups of a group G.
Determine with proof whether each of the following is a subgroup of G.
(a) �
α∈A Hα
(b) �
α∈A Hα
Suppose H is a subgroup of G and H ⊆ H1 ⊆ G ⊆ G1. We can make the
following observations about relationships between these sets. First, if H1 is a
group under the same binary operation, then the fact that H is a subgroup of G
implies H is a subgroup of H1 also. For if H exhibits properties H1–H3 as a subset
of G, it also does so as a subset of H1. Similarly if G1 is a group, then H is a
subgroup of G1. The most efficient way to describe this latter relationship is to say
that < is transitive: If H