Math 330: Algebra III

Pictures

Cayley Digraphs The cyclic group $$Z7 The dihedral group $$D14 The lattice $$Z2 = Z(+)Z The abelian group $$Z12= Z3 (+) Z4. The abelian group $$Z132= Z11 (+) Z12, and the torus $$T2= S1 x S1 The quaternion group $$Q8. The permutation group $$S4. The tetrahedral group. The free group on two generators.

Number Theory GCDs as `intersections' and LCMs as `unions' The integers arranged in a lattice of divisibility The Euclidean algorithm. Complex Numbers Cartesian form and polar form for complex numbers Complex addition is just 2D `vector' addition. Complex multiplication, in polar form. The complex exponential map. Isomorphism Theorems The Fundamental Isomorphism Theorem. The Diamond Isomorphism Theorem. The Chain Isomorphism Theorem. The Lattice Isomorphism Theorem.

Morphisms An isomorphism of $$Z/12 with $$Z/4(+)Z/3. An automorphism of $$Z/7. An epimorphism from the real line to the unit circle. An epimorphism from $$Z/12 to $$Z/6. A monomorphism from $$Z/6 to $$Z/12. The projection from Z to $$Z/3. The exponential map from R to C. Subgroups Cayley digraphs of $$Z/7 with 6 different generators. The lattice of subgroups of $$Z/6. The lattice of subgroups of the group $($Z,  +) The subgroup generated by $(2,0)$ and $(1,2)$ in the group $$Z (+) Z.

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This page last updated: 2002-11-04