** Week I**: Find a binary relation that is
** symmetric **, ** transitive **, but not
** reflexive**, or show that one doesn't exist.

** Week II**: Is the symmetry group of the cube a dihedral group?

** Week III(a)**: Show that the symmetry group of the cube
is also the symmetry group of the octahedron.

** Week III(b)**: What's the largest order of any element of the symmetric
group S_26?

** Week III(c)**: Prove that the following group is the trivial
group: G = (x,y,z | zxyzzyyx, zxyz, xyy)

** Week IV**: Find a group G with the following property: All the proper subgroups of G
are cyclic, but G is not cyclic itself.

** Week VII**: Let G be the group of symmetries of
the icohadedron. Construct a perumation representation of
G on the five perfect cubes that can be inscribed inside the
icosahedron such that the vertices of the cube are vertices
of the Icosahedron (If you can't visualize how to inscribe a
cube inside an icosahedron,
look here).
What is the image of this permutation
representation? What is the kernel?

** Turkey Challenge**: Let G be a group with M distinct conjugacy classes and
let H be a subgroup with N distinct conjugacy classes. Show by example that it is
possible that N > M. Does there exist an example when G = S_n for some n?

** Week IX**: Still in preparation.

fcale@math.northwestern.edu

Copyright ©
1999 - 2006 Michael R. Stein