S
UMMARY OF
9/24 L
ECTURE
We started lecture by talking more about
Z
n
, a cyclic group of order
n
which is generated by
the element
1
+
subject to the condition
n
≡
0 (
mod
n
)
. In other words,
Z
n
=
{
0
,
1
,
2
, . . . , n

1
}
under the binary operation ‘addition
(
mod
n
)
’: just like ordinary addition except that
n
is
the same thing as 0, which we write in the form
n
≡
0 (
mod
n
)
. From this it follows that
n
+ 1
≡
1 (
mod
n
)
,
n
+ 5
≡
5 (
mod
n
)
, etc.
A natural question is, if we can do addition
(
mod
n
)
on
Z
n
, how about multiplication
(
mod
n
)
?
In other words, is
Z
n
a group under multiplication
(
mod
n
)
? Well, it’s certainly closed, asso
ciative, and has an identity – namely, 1. But, not every element is an inverse... as usual, 0 isn’t.
(Because 0 times anything is 0, and therefore will never give the identity.)
OK, so we remove 0 from
Z
n
. Now do we have a group under multiplication? After some
playing around, we determined that the answer is, sometimes. The way we played around with
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview.
Sign up
to
access the rest of the document.
 Fall '10
 GideonMaschler
 Addition, Multiplication, Ring, Prime number

Click to edit the document details