Abstract Algebra summary

1. Recap on sets, functions and relations
   1. Standard notation
   2. Functions; restriction, associativity of composition
   3. Images and pre-images; properties
   4. Checking a function is well-defined
   5. Equivalence relations and classes; partitioning
   6. Introduction to groups
2. Permutation groups
   1. Permutations; grouphood of Sym(S)
   2. Cycle decomposition (existence and uniqueness); orbits
   3. Examples. Isomorphism of D₃ to S₃
   4. Order of a permutation; order as LCM of cycle lengths
   5. Parity, with well-definedness; elements of A₇
   6. Conjugates of permutations. Conjugacy Û same type.
3. Groups
   1. Definitions and notation
   2. Basic consequences of axioms:
      • Uniqeness of identity and inverse
      • Generators, cyclic groups, order of elements
      • g_r=e Þ o(g)|r
      • Two facts about finite cyclic groups
      • Conjugate elements and conjugacy classes. Example: D₆
   3. Isomorphisms
      • Preservation of identity and inverse
      • Isomorphism of cyclic groups to Z
   4. Creating new groups. Intersections
   5. Subgroup lattices and partial orderings
4. Cyclic groups
   1. Basic theory. Isomorphism to Z, cyclicity of subgroups
   2. Infinite cyclic groups; adding same. Prime multiplication groups
   3. Finite cyclic groups. Properties. _pq @ _p × _q
5. Lagrange's theorem
   1. Statement. Consequence (prime order groups are cyclic)
   2. Cosets
   3. Properties of cosets: other defn., partitioning, bijection
   4. groups of order £ 7. 4 and 6 in gory detail.
6. Normal subgroups and quotient groups
   1. Normal subgroups. Equivalent definitions. H G if G contains 2 cosets of H.
   2. Quotient groups
   3. Examples of same
7. Homomorphisms
   1. Basic facts. Kernel (is a normal sg, and converse) and image (sg)
   2. Examples
   3. Isomorphism theorem
   4. Examples: Aut(S₃) @ S₃, homs from S₄ to A₄, S₃
   5. Quotient groups via congruences
   6. Epilogue

This document was translated from L^AT_EX by H^EV^EA.