2.2. SUBGROUPS AND CYCLIC GROUPS
95
denoted
h
S
i
. When
S
D f
a
g
is a singleton, the subgroup generated by
S
is denoted by
h
a
i
. We say that
G
is generated by
S
or that
S
generates
G
if
G
D h
S
i
.
A “constructive” view of
h
S
i
is that it consists of all possible prod
ucts
g
1
g
2
±±±
g
n
, where
g
i
2
S
or
g
±
1
i
2
S
. Another view of
h
S
i
, which
is sometimes useful, is that it is the intersection of the family of all sub
groups of
G
that contain
S
; this family is nonempty since
G
itself is such
a subgroup.
The family of subgroups of a group
G
are partially ordered by set
inclusion.
1
In fact, the family of subgroups forms what is called a
lattice
.
2
This means that, given two subgroups
A
and
B
of
G
there is a unique
smallest subgroup
C
such that
C
²
A
and
C
²
B
. In fact, the subgroup
C
is
h
A
[
B
i
. Furthermore, there is a unique largest subgroup
D
such that
D
³
A
and
D
³
B
; in fact,
D
D
A
\
B
.
Cyclic Groups and Cyclic Subgroups
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This note was uploaded on 12/15/2011 for the course MAC 1105 taught by Professor Everage during the Fall '08 term at FSU.
 Fall '08
 EVERAGE
 Algebra

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