Cumulative List of Topics

Here I will list topics covered in class, along with page numbers in the textbook.  So far, these have mainly concerned Chapters 1-3 (up to page 84).  However we are about to begin Chapters 4-6, so by now you should have read at least up to p.142.  I will try to keep this list updated as we progress through the material covered.

• Informal definition and first examples, e.g.
  • addition in number systems p.20 including the integers p.22, reals, rationals, complex numbers, integers mod n, p.21;
  • multiplication in invertible matrices and nonzero reals, more generally GL_n(F) = GL(n,F)  p.29;
  • special linear group SL_n(F) = SL(n,F)  p.29;
  • symmetry groups of objects and patterns, e.g. the symmetry groups of squares, cubes, etc.).  Dihedral groups p.6.
  • Klein four-group.  p.59
  • Quaternion group Q₈ of order 8.  p.57 
• Order of a group p.40, denoted |G|.  Order of a group element p.48, denoted |g| = ord(g) = o(g).
• Cayley tables, including addition and multiplication tables.  p.7
• Finite and infinite groups p.40.
• Binary operations p.38.  Formal (axiomatic) description of groups.  p.38
• Cycle notation for permutations p.15.  The symmetric group S_n and, more generally, Sym(X) (which the book calls Perm(X)) p.11.
• Subgroups of G, p.61
  • The subgroup 〈S〉 generated by a subset S of G.  p.63
  • The cyclic subgroup 〈g〉 generated by an element g in G.  p.63
  • Testing to see if a subset is a subgroup.  p.62
  • Hasse diagrams for subgroups of a group.  p.168
  • The center of G, denoted Z(G).  p.64
  • The centralizer of a subset S ⊆ G, denoted C_G(S).  p.64
• Isomorphisms from G to H; isomorphic groups G ≅ H.  p.53
• Permutation groups (subgroups of S_n).  p.81
  • Cayley's Representation Theorem: Every finite group is isomorphic to a permutation group,  p.225  We have not formally proved this but we presented the idea behind the proof.
• Transpositions (i j) as generators for S_n.  p.76
• Even and odd permutations, and the alternating group A_n.  p.78
• Direct product of groups G × H.  p.59
• Lagrange's Theorem: For finite groups, if H ≤ G then |H| divides |G|.   p.115.  We have not yet proved this theorem!  But we have proved some special cases, starting with the case of finite cyclic groups.
  • Corollary: The order of each element of a finite group, divides the order of the group.  p.115.  So far, we have proved this only in the abelian case.
• For a group G acting on a set X:
  • The orbit O(x) = O_G(x) of a point x in X.  p.98
  • The stabilizer G_x = Stab_G(x) of a point x in X.  p.93
  • The orbit-stabilizer formula |O(x)|=[G:G_x], i.e.  |G|=|G_x||O(x)|.  p.130
  • Special case: G acts on G by conjugation,  p.100.  The orbits are conjugacy classes; the stabilizers are the centralizers.  Hence the number of conjugates of g in G is [G : C_G(g)].
• Left vs. right cosets, p.110.  Normal subgroups, p.188
• Simple group.  p.202.  The Classification of Finite Simple Groups (CFSG), p.202
• Solvable group, p.206.  To fully appreciate the meaning of these terms requires understanding quotient groups G/K, which unfortunately we haven't covered, p.198.
• Group homomorphisms, p.210.  The kernel of a homomorphism from G to H is a normal subgroup of G.  Automorphisms.  p.210

/ revised December, 2023