MAT301H1: Groups and Symmetries
•
In
V
4
={
e ,a ,b ,a b
}
,
∣
e
∣=
1 ,
a
⋅
a
=
e
⇒∣
a
∣=
2 ,
b
⋅
b
=
e
⇒∣
b
∣=
2 ,
∣
a b
∣=
2 .
•
In
Q
8
={
1,
−
1,
i ,
−
i , j ,
−
j ,k ,
−
k
}
,
∣
1
∣=
1 ,
∣−
1
∣=
2 ,
∣
i
∣=∣
j
∣=∣
k
∣=
4 .
Definition: Subgroup
H
is a subgroup of
G
iff
H
is a subset of
G
which is a group under the same operation as
G
.
Example
In
D
4
,
{
e ,r ,r
2,
r
3
}
is a subgroup since it is closed under × (
r
n
×
r
m
=
r
n
m
), has inverses (
r
−
1
=
r
3
,
r
2
−
1
=
r
2
), and
e
∈{
e ,r ,r
2,
r
3
}
.
Definition: Proper Subgroup
H
is a proper subgroup of
G
iff
H
is a subgroup of
G
and
H
≠
G
,
H
≠{
e
}
.
Note
Does
G
always have a proper subgroup? No.
{
e ,r ,r
2
}
has no proper subgroups.
Definition: Set of Generators
S
is a set of generators of a group
G
iff every
g
∈
G
can be expressed as:
•
multiplication:
g
=
s
1
m
1
×
s
2
m
2
×⋯×
s
k
m
k
•
addition:
g
=
m
1
s
1
m
2
s
2
⋯
m
k
s
k
where
m
i
∈ℤ
and
s
i
∈
S
(repetitions of
s
i
's are allowed). Any such combinations of