Abstract Algebra IBL Wiki

• Why we need associative

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.

• Simple Shift on 26 letters - Introduces Modular arithmetic
• The Order of a Simple Shift - Introduces the fact that the order of an element divides the order of a group. This comes from
• Lagrange's Theorem
• group homomorphism, group isomorphism, kernel of a group homomorphism, image of a group homomorphism
• Generator of a Group

• Cayley Graph, generating set, presentation
• Cayley Subgraph, Hasse Diagram
• Collapsible Subgraph, normal subgroup, Coset
• External Direct Product, Internal Direct Product, Direct Sum, Product of Group Subsets
• factor group, quotient group
• center of a group, centralizer of an element
• normalizer of a subgroup, centralizer of a subgroup

• Soccerballahedron
• Cayley Graph of Z direct sum Z

• Exploring the set products HK and KH on D4

• Nim

• Fundamental Theorem of Finitely Generated Cyclic Groups
• Fundamental Theorem of Finitely Generated Abelian Groups
• First Isomorphism Theorem

• Group
• Subgroup
• Permutation Group
• Symmetric Group

definition:group_homomorphism