The order of a group element is the least positive integer such that , where is the identity element; its order is infinite if no such exists.
Let have finite order . A group homomorphism preserves the group operation and the identity, so
The order of an element divides every positive exponent that gives the identity. Hence
If is a surjective function and has order , choose with . The first result gives , where . The element
then has order , because the order of is .
A group homomorphism is determined by the image of a generator of the cyclic group , and that image must have order dividing . In the symmetric group , the only such elements are the identity and the three-cycles. There are
three-cycles. Therefore the number of homomorphisms is
On the part with , write
This is the upper portion of a hyperboloid of one sheet. Choosing the outward orientation, whose normal points radially away from the -axis, the vector area element is
Its radial component is positive and its vertical component is negative. Reversing the normal reverses all the fluxes below.
For
the curl is
On the surface, , and therefore
Direct integration gives
To verify this with the Stokes theorem, note that on a circle of fixed ,
The induced boundary orientation runs in the negative direction on the top circle and the positive direction on the bottom circle. Hence
For , Stokes' theorem again reduces the flux to its two boundary circles. The outward orientation gives positive direction at , where , and negative direction at , where . Thus
Expanding around the expected value ,
because . Therefore
with equality exactly when .
For the absolute loss,
Leibniz differentiation gives, with ,
Thus decreases while and increases while . It is minimized at any median, characterized in the continuous case by

Pinned article: Introduction to the OurBigBook Project

Welcome to the OurBigBook Project! Our goal is to create the perfect publishing platform for STEM subjects, and get university-level students to write the best free STEM tutorials ever.
Everyone is welcome to create an account and play with the site: ourbigbook.com/go/register. We belive that students themselves can write amazing tutorials, but teachers are welcome too. You can write about anything you want, it doesn't have to be STEM or even educational. Silly test content is very welcome and you won't be penalized in any way. Just keep it legal!
We have two killer features:
  1. topics: topics group articles by different users with the same title, e.g. here is the topic for the "Fundamental Theorem of Calculus" ourbigbook.com/go/topic/fundamental-theorem-of-calculus
    Articles of different users are sorted by upvote within each article page. This feature is a bit like:
    • a Wikipedia where each user can have their own version of each article
    • a Q&A website like Stack Overflow, where multiple people can give their views on a given topic, and the best ones are sorted by upvote. Except you don't need to wait for someone to ask first, and any topic goes, no matter how narrow or broad
    This feature makes it possible for readers to find better explanations of any topic created by other writers. And it allows writers to create an explanation in a place that readers might actually find it.
    Figure 1.
    Screenshot of the "Derivative" topic page
    . View it live at: ourbigbook.com/go/topic/derivative
  2. local editing: you can store all your personal knowledge base content locally in a plaintext markup format that can be edited locally and published either:
    This way you can be sure that even if OurBigBook.com were to go down one day (which we have no plans to do as it is quite cheap to host!), your content will still be perfectly readable as a static site.
    Figure 5. . You can also edit articles on the Web editor without installing anything locally.
    Video 3.
    Edit locally and publish demo
    . Source. This shows editing OurBigBook Markup and publishing it using the Visual Studio Code extension.
  3. https://raw.githubusercontent.com/ourbigbook/ourbigbook-media/master/feature/x/hilbert-space-arrow.png
  4. Infinitely deep tables of contents:
    Figure 6.
    Dynamic article tree with infinitely deep table of contents
    .
    Descendant pages can also show up as toplevel e.g.: ourbigbook.com/cirosantilli/chordate-subclade
All our software is open source and hosted at: github.com/ourbigbook/ourbigbook
Further documentation can be found at: docs.ourbigbook.com
Feel free to reach our to us for any help or suggestions: docs.ourbigbook.com/#contact