- Each part of this problem might be useful for the next!
- (a) Let Z be the center of a group G. If G/Z is cyclic, show that G is abelian (and hence G=Z).
- (b) If G is a nonabelian group of order 10, show that the class equation for G must be 10 = 1 + 2 + 2 + 5.
- (c) If G is a nonabelian group of order 10, show that G is
isomorphic to D
_{5}. [Hint: conclude from (b) that G has an element x of order 5 and an element y of order 2. Show that yxy = x^{4}, and show that this uniquely determines the multiplication table of G. Hence, up to isomorphism, there is a unique nonabelian group of order 10.]

- Prove Proposition 1.12 on p. 199.
- Let G be the group of order preserving symmetries of
the set X = {(a,0,1), (a,0,-1), (-a,1,0), (-a,-1,0)}, where
a = 2
^{-1/2}. Note that X is the set of vertices of a regular tetrahedron T centered at the origin. Thus G is the tetrahedral subgroup of SO_{3}. Determine the set S of poles of G (describe them explicitly, either in terms of T or by giving their coordinates), determine |S|, and explicitly determine the orbits of G acting on S (i.e., which poles are in which orbits). You do not need to justify your answers (except to yourself!). - (Burnside's formula)
Let G = {g
_{1}, ..., g_{r}} be a group of order r acting on a finite set S. Let n be the number of orbits. For each g in G, let S^{g}= {s in S : gs = s}. Prove that n|G| = |S^{g1}| + ... + |S^{gr}|. [Hint: Look at C = {(g,s) : g is in G, s is in S, gs = s}.] - (A generalization of Cayley's theorem)
Let H be a subgroup of a group G of finite index [G:H] = n.
Let G act on the left cosets G/H of H in G by left multiplication.
Show there exists a homomorphism f: G -> S
_{n}such that ker f is contained in H and every normal subgroup of G which is contained in H is contained in ker f (i.e., ker f is the largest normal subgroup of G contained in H). - Let L = {me
_{1}+ ne_{2}: m and n are integers}. Note that L is an additive subgroup of**R**^{2}. Let v = (0.1, 0.01)^{t}and let G = Sym(v + L) be the rigid motions of the plane which preserve the coset v + L.- Determine the point group of G (actually list the matrices explicitly; no proof is needed).
- Determine the translation subgroup L
_{G}of G (no proof is needed, but be clear as to which vectors are in L_{G}). - Artin shows that the point group acts on L
_{G}(see p. 169), but goes on to say that ``it is important to note that ... G need not operate on L_{G}.'' Verify this by exhibiting an element g of G and a vector w in L_{G}, such that gw is not in L_{G}.