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In the early 30's, Germany was in economic ruin. In that context, desperate people would flocked to any leader who promised a way out. And if he blamed the ills of their nation on a small minority, so much the better. "Not our fault we lost the war, not our fault our currency has the value of toilet paper, it's those damn Jews!" And history unfolds from there. But we're not in economic ruin. We haven't been roundly defeated in a World War. We've gott some problems sure, but no more than we had ten years ago, and maybe less, as measured by some indicators such as unemployment. So what possible excuse can we offer for supporting someone as Vile as Trump? {s/Jews/Muslims/g S/Poles/Mexicans/g …} Referring to gitmo, a facility that has tortured innocent people (bycatch) to the point of suicide in its patriotic fervor, Trump vows, "I would keep it open and load it up with bad dudes." And he smiles - smiles when he thinks of torturing others. You can hear the joy in his voice as he elaborates, "They're going to tell us everything they know." Some might say Gitmo is a regretable business that must be done in the interest of national security and the greater good, but Trump thinks it's fun! His love for violence was on display again when he encouraged security to "knock the crap" out of a protester. "I will pay for your legal fees, I promise. You know what they used to do to a guy like that? He’d be carried out on a stretcher." If my blakc son had made such a threat in public, and on video, he'd be in prison facing felony charges - but that's another story. Trump's personality is compatible with the worst dictators in human history. all he needs is some technology and a loyal following. How is this possible in the United States of America? Why does this monster have even one supporter? I think the media plays a role, with their unrelenting fear mongering, but there's probably more to it than that. In any case, it's terribly frightening that he might get elected to anything higher than dog catcher, and personally, I'm not sure how he would treat the dogs - we know he doesn't like creatures who allow themselves to be captured. Must Read

Welcome to the Math Reference project, an electronic archive of mathematical information. Topics range from high school geometry up to graduate level topology, and (almost) everything in between. All material is copyright Karl Dahlke, 2002-2015, and may not be copied or redistributed without permission.

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To fill this void, I am writing a textbook based upon the material in these pages. It guides the reader through a wide array of topics in abstract algebra in a specified order. Each chapter builds on the information that has gone before, and forward references are rare, though they do happen from time to time. I hope this on-line book is more accessible than a sea of disconnected web pages. In fact, I am looking forward to the day when it serves as the basis for a course at the undergraduate or graduate level.

The book is not yet complete, but you are welcome to read it, and provide feedback, and even participate in its development if you wish.

As I reformulate my material for this project, I am fixing more than a few errors, and making things clearer, with more examples etc. I can't do all this work twice, so please understand if I occasionally direct you to a chapter in the book, rather than a page in the MathReference tree. Some of this redirection is done by http code 302, which is the best way to bring search engines up to date, but could be confusing if you have already bookmarked a page on this website, and you suddenly find yourself somewhere else. I apologize for any inconvenience.

This website assumes you know the basics, i.e. high school algebra and geometry. If you're patient, and you have some innate talents, a solid high school background is enough to begin most topics. However, you probably can't descend into any of these topics in detail until you've covered most of them at a high level. If you're just starting out, you sort of have to learn math breadth first, rather than depth first, since areas of math tend to call upon one another as they get more advanced.

For now, all information is in html. This means I don't use some of the strange symbols commonly employed in higher mathematics. Fortunately, we won't need those for a while, and in the meantime this site remains accessible to all. Even a blind user should be able to access this site using edbrowse.

Although I am avoiding, or at least postponing, some of the more complex notation, we will still need some basic symbols such as × and ÷, and the Greek alphabet. I am using the unicode standard to represent these symbols - I hope your browser is compatible.

I try to use the times symbol × for arithmetic multiplication, and * for generic multiplication in a group or ring. On rare occasions, × denotes ordinal or cardinal multiplication, a natural extension of integer multiplication. When applied to modules, u×v is the tensor product of u and v. When * is applied to sets it indicates their cross product, or their direct product if you prefer.

The symbol "/" and the word "over" both indicate division, but over is a lower precedence operator, and is usually used for fractions. Thus (a+b)/(c+d) is the same as a+b over c+d. I try to put spaces around the lower precedence operators such as = and over, to make the equations a little clearer.

Of course / means many more things, including quotient group, quotient ring, quotient module, fraction ring, quotient space, field extension, integral extension, and so on. It is the most overloaded of all the operators. Sorry about that.

If f and g are functions, I will sometimes write fg for f followed by g. Of course, if parentheses are used, they rule the day. Thus fg applied to x is g(f(x)). In other contexts, fg might mean g followed by f, but this is not the typical interpretation of juxtaposition (on my website), so I'll try to be very explicit when fg means g compose f.

When I say "assume" x = 3, then x must equal 3 for this theorem, or this part of the argument, to hold. If I say "suppose" x = 3, this is a proof by contradiction, and I will soon show that x cannnot equal 3.

Variables that stand for numbers, or points, or functions, are in lower case. Variables that represent sets are in upper case. So you might see: let x be an element of the group G. I don't usually write g an element of the group G, as that is simply too confusing for my blind subscribers. (This illustrates why one cannot simply "transcribe" a proof, to render it accessible.) However, I may let g1 and g2 be two elements of G, whence g1*g2 is also in G.

Bold text is reserved for fixed mathematical structures, like the reals **R**,
or the rationals **Q**, or the alternating group on 9 letters **A**9,
or the n sphere **S**n.
Thus the loops in a topological space S are the continuous maps from **S**1 into S.

You'll notice that each page in this archive begins with a light blue navigation panel. The down arrows take you to subtopics, areas of math that are "spin-offs" or specializations of the current topic. The up arrows take you back to higher level topics. There are no up arrows here, since this is the main page. You might follow down arrow links to "calculus", then to "arc length and curvature", then take the up arrows back to calculus, then back to this page. If you are used to directory structures (Unix), or folders within folders (Windows), this layout should seem natural. A topic is like a folder, and the web pages within that topic are the files in the folder. They explain the topic in detail, and should be read sequentially, starting with the introduction. you can step forward and backward through the pages of a given topic using the left and right arrows at the bottom of each page. Thus you need not retrieve the entire archive on integral calculus just to review integration by parts. This saves time for the folks with dial-up connections, and if you're doing a keyword search, you see only the information you requested. However, it that information is unclear when taken out of context, you may need to back up a couple of pages, or retreat to the introductory page and start from there. The navigation panel always provides a link to the start of the current topic, i.e. the introductory page.

Topics below the current topic are only accessible from the introductory page. If you're reading through that topic, well, I figure you don't want to descend, so why clutter the pages with additional links? You can always jump back to the intro page, then descend to a lower level. Two clicks instead of one; that's not bad.

The pages within a topic are also called sections. This is less ambiguous, for a web page, when printed out, is often several pages long. So I may refer you to "the previous section", or "an earlier section" in the current topic, or perhaps "another section" in a different topic. These references usually include direct links, so you don't have to scroll down and use the arrows.

When you use the search function, don't bother with punctuation; I throw it all out. There is no "advanced search". Try to use words, rather than ascii equations. Ask for "pythagorean triples", rather than "integers x^2 + y^2 = z^2".

This website requires javascript, but it does not employ cookies or plugins. The javascript is minimal, and is supported by virtually every browser.

Ted D.

Just wanted to tell you that I think your math reference is great!!
Your reference has been immensely helpful to me.
It's super-readable, and somehow the proofs tend to be at just the right level of detail.
What I liked most about your proofs is
how much of them are in plain English.
The proofs are concise and feel somehow morally correct
rather than merely technically or mechanically correct.
It's a great resource.

Andrew H.

I wish to thank you very very much for the time that you afforded me.
I really appreciate and understand your excellent presentation.

Dan B.

First, let me say that MathReference.com is amazing!
I teach high school mathematics and have just started work on a master's degree
in pure mathematics, and have been seeking out additional explanations or
descriptions of some of the concepts in my graduate abstract algebra course.
You have done a great job fleshing out
these concepts (such as a normal subgroup, quotient group, etc.).
In short, I can't thank you enough for the assistance!
I'm recommending your site to other graduate students.

Shelley W.

I am a math tutor, and I just want to let you know that
your site is Fantastic...very user friendly:
logically organised, easy to navigate and simply explained.
Bravo!

Barry R.

I am currently creating a spreadsheet to determine the partial surface area of
spheroid tanks.
Your website has been a big help as I have had to give myself a refresher course in integrals.

Mary O.

I just used your math site for my 1st year engineering students.
They were getting a bit confused as to the impact of elementary
row operations on the determinant of a matrix so I googled .
Your site covered exactly what we needed and also opened up another option for
the little darlings to solve their own problems.
Thanks.

Jens G.

I would like to thank you for having put your site on the web.
For me it is often the quickest online reference in case I'd like to know more
about a certain topic than just some theorems without proof or introduction.
Good work!

Ben B.

I found your site the other week, very nice,
i like that its so easy to understand.

Vince C.

I am a real fan of your website,
it serves a real need by addressing mathematical issues that are often poorly,
or incorrectly, presented in many standard texts.
Although not a "licensed" mathematician, I am fairly math-literate,
and sometimes I think your notation may be a bit "compressed"
for lower level students, but that is strictly my opinion,
and I'm aware that notation is frequently a fine line.

Rashida J.

Your site is wonderfull.
Lots of information and in very simple language.
Thank you for doing it and sharing it.

Michael Q.

I want to thank you very much for your remarkable mathematics data base.
I discovered it today when I wanted to refresh my memory about the
Schroder-Bernstein theorem.
It was very helpful for me.
I also glanced with interest and admiration at other parts of your website.

Harry H.

I just thought I'd let you know I think your site is excellent.

David M.

Great site, finally the advanced stuff in English!!

Jeff B.

Thanks for the articulate restatement of the dihedral group.
(It was 25 years ago, and I'd forgotten this example.)

Hiral P.

Thanks a lot for your help.
I have looked up so many topics from your website, and I must say its very helpful.

Francois C.

Congratulations for this site!
I found it when googling for `external semidirect product' and was happy to
find - at last! - a site where common and useful concepts are clearly explained.
Up to now, I had to compile myself some `Working Notes',
just to avoid having to search through a thick jungle of elementary literature…
Thank you also for not using the `classical' hieroglyphs of math (although it
might help sometimes, if only because these are customarily used throughout the
literature), so that any browser can display a legible version of the text.
I must encourage you, and be sure I will recommend your site to my colleagues.

Barry H.

I enjoy lots of your articles…those that I can understand.
I learn a lot about things that I never knew existed.

Juha V.

I stumbled upon mathreference while searching for Hahn-Banach theorem,
but I ended up finding much more.
Thanks.

Geoffrey R.

Thank you again for answering my questions -- and thank you for doing so in
a way a Master's degree student can understand.
It is rather difficult for me to try to understand an explanation written, literally, in Greek (or any
other foreign language).

Eric B.

I just discovered your web site this morning and found it helped me at once find
information about the "geometric distribution".
Super helpful!
Thank you!