S
UMMARY OF
11/3 L
ECTURE
We reviewed a couple of notions. Most importantly, recall that the order of an element
g
of a
group
G
is
deﬁned
to be the order of the cyclic subgroup generated by that element:

g

=
h
g
i
.
Another way to think about

g

is that it is the smallest positive integer
n
such that
g
n
= 1
(i.e.
g
6
= 1
,g
2
6
= 1
,g
3
6
= 1
,...,g
n

1
6
= 1
, but
g
n
= 1
). It’s a good exercise to ﬁgure out why the
order satisﬁes this property. (Also, note that
g,g
2
,...,g
n
are all distinct!)
Next, we discussed direct products. Given two
sets
A
and
B
we deﬁne a new set
A
×
B
, called
the direct product
of
A
and
B
, to be the collection of all ordered pairs of elements, the ﬁrst
from
A
, the second from
B
. In mathish,
A
×
B
=
{
(
a,b
) :
a
∈
A,b
∈
B
}
.
For example,
R
×
Z
is the set of all ordered pairs of the form
(
x,n
)
where
x
is any real number
and
n
is any integer. Thus,
(3
.
14159
,

7)
∈
R
×
Z
, but
(3
,
2
.
1)
6∈
R
×
Z
.
We then concluded class by going over Cauchy’s theorem once more, this time completing the