Open main menu
Home
Random
Recent changes
Special pages
Community portal
Preferences
About Wikipedia
Disclaimers
Incubator escapee wiki
Search
User menu
Talk
Dark mode
Contributions
Create account
Log in
Editing
Quotient group
(section)
Warning:
You are not logged in. Your IP address will be publicly visible if you make any edits. If you
log in
or
create an account
, your edits will be attributed to your username, along with other benefits.
Anti-spam check. Do
not
fill this in!
=== Remainders of integer division === A slight generalization of the last example. Once again consider the group of integers <math>\Z</math> under addition. Let {{tmath|1= n }} be any positive integer. We will consider the subgroup <math>n\Z</math> of <math>\Z</math> consisting of all multiples of {{tmath|1= n }}. Once again <math>n\Z</math> is normal in <math>\Z</math> because <math>\Z</math> is abelian. The cosets are the collection {{tmath|1= \left\{n\Z, 1+n\Z, \; \ldots, (n-2)+n\Z, (n-1)+n\Z \right\} }}. An integer <math>k</math> belongs to the coset {{tmath|1= r+n\Z }}, where <math>r</math> is the remainder when dividing <math>k</math> by {{tmath|1= n }}. The quotient <math>\Z\,/\,n\Z</math> can be thought of as the group of "remainders" modulo {{tmath|1= n }}. This is a [[cyclic group]] of order {{tmath|1= n }}.
Edit summary
(Briefly describe your changes)
By publishing changes, you agree to the
Terms of Use
, and you irrevocably agree to release your contribution under the
CC BY-SA 4.0 License
and the
GFDL
. You agree that a hyperlink or URL is sufficient attribution under the Creative Commons license.
Cancel
Editing help
(opens in new window)