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--- ben_s_outline [2013/08/14 10:07]
bmwoodruff
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-I plan to have multiple threads running simultaneously through the semester.  However, for development purposes, it will probably be easiest to organize them into sections.  I'm hoping to grab a lot of ideas about combinatorial games from Dana.  I'm not sure what you are planning Josh, but I hope to see it soon.
-=== Miscellaneous ===
-  *[[Problem:Why we need associative]]
-=== Cryptography ===
-We'll start with a simple cryptosystem that involves shifting. Then advance to matrix encryption, then matrix modular encryption.  This gets to GL(n,m) really fast, and introduces homomorphisms via the determinant map, and isomorphisms between U(p-1) and Z(p) for prime p. I want to get to homomorphisms quickly, as I want to introduce the concepts of factor groups, normal groups, cosets, and more, from the context of homomorphisms.
-  *[[Problem:Simple Shift on 26 letters]] - Introduces Modular arithmetic
-  *[[Problem:The Order of a Simple Shift ]] - Introduces the fact that the order of an element divides the order of a group.  This comes from
-  *[[theorem:Lagrange's Theorem]]
-  *[[definition:group homomorphism]], [[definition:group isomorphism]],
-  *[[definition:Generator of a Group]]
-=== Cayley Graphs ===
-  *[[definition:Cayley Graph]], [[definition:generating set]], [[definition:presentation]]
-  *[[definition:Cayley Subgraph]], [[definition:Hasse Diagram]]
-  *[[definition:Collapsible Subgraph]], [[definition:normal subgroup]], [[definition:Coset]]
-  *[[definition:External Direct Product]], [[definition:Internal Direct Product]], [[definition:Direct Sum]], [[definition:Product of Group Subsets]]
-  *[[definition:factor group]], [[definition:quotient group]]
-=== Combinatorial Games ===
-  *[[definition:Nim]]
-  *
-=== Fundamental Theorems ===
-  *[[theorem:Fundamental Theorem of Finitely Generated Cyclic Groups]]
-  *[[theorem:Fundamental Theorem of Finitely Generated Abelian Groups]]
-  *[[theorem:First Isomorphism Theorem]]
-=== Definitions ===
-  *[[definition:Group]]

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