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.
Copyright © 1999 - 2006 Michael R. Stein