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Are there any books that present theorems as problems? To be more specific, a book on elementary group theory might have written: "Theorem: Each group has exactly one identity" and then show a proof or leave it as an exercise. The type of book that I am imagining would have written "Problem: How many unit elements can a group have?" and similarly for all other theorems.

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    $\begingroup$ Why in the world would you look for such a book? Are you going to use it as a torture device on unwitting undergrads? $\endgroup$
    – user577
    Jan 23, 2010 at 2:52
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    $\begingroup$ Incidentally, "zero" is a lousy name for the identity element in a group. Almost all groups in nature are essentially multiplicative --- they arise as matrix groups --- whence 0 is not a group element. The best name is "unit". Actually, given that this is a CW question, I feel no compunction about changing it myself :) $\endgroup$ Jan 23, 2010 at 17:33
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    $\begingroup$ Unit is bad too. Identity or unity are fine, but unity is typically also reserved for rings. The problem is, once you go from group theory to ring theory, and you encounter the proper usage of unit, it'll mess everything up. I mean, by the definition of a unit, every element of a group is a unit. $\endgroup$ Jan 23, 2010 at 18:53
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    $\begingroup$ This is known in certain circles as the "Moore Method" ... famously (or infamously) practiced by R. L. Moore. LINK: mathoverflow.net/questions/12070/… $\endgroup$ May 19, 2010 at 14:35
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    $\begingroup$ (cont.): I'm not so sure that the weeding argument above is on the mark... $\endgroup$
    – Jon Bannon
    May 14, 2012 at 23:10

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Whyburn and Duda, Dynamic Topology.

(Whyburn was a student of Moore, as in the Moore Method mentioned in the comments above.)

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Not exactly a book, but my web page displays a list of exercises in Matrix theory. Among the 440 exercises, about a half can be viewed as proving theorems.

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  • $\begingroup$ +1. Little bugreport: on page 1, "exercises.pdf" should be "exo.pdf". Also, if you use the hyperref package and the \href and \url commands, the links will become clickable. $\endgroup$ Jul 3, 2015 at 8:49
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Many people are work on various variations of Inquiry-Based Learning (in math often called the Moore Method). For instance, I have a book for Introduction to Proofs. For lots of excellent examples check out the Journal of Inquiry-Based Learning in Mathematics.

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Théorie des groupes is an undergraduate book in Group Theory answering the question... Unfortunally (or fortunatelly depending on who you are) written in French. A very good book on the topic, including many examples of groups,Sylow theorems, nilpotent groups...

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