M901 Assignment 1

Instructions: Do any three of the following problems. This problem set is due Monday Sep 8.

The following definition will be useful for problems 1 and 2:

Definition: Let g:S -> G be a mapping where S is a set and G is a group. We say (G,g) is the free group on (or generated by) the set S if given any map h:S -> H of S to a group H, there is a unique homomorphism f:G -> H such that fg=h.

• (a) If (G,g) is free on a set S, prove g is injective.
• (b) Let S= {s} be a singleton set. Let G be the integers under addition. Define g:S -> G by g(s) = 1. Prove that (G,g) is the free group on the set S.
1. Let S be any set. Let F={Gs}s in S be a disjoint family of groups with an isomorphism fs:Z -> Gs for each s in S. Let G be the free product over the family F (i.e., G is the categorical sum, in the category of groups, of the groups in the family; I'm not using the notation from class since it's hard to do it on the web). Define f:S -> G by f(s)=isfs(1) for each s in S, where is: G_s -> G is the canonical inclusion. Show that (G,f) is free on the set S. (This shows that free groups are special cases of free products; the group that's free on a set S is the free product of copies of the integers, one copy per element of S.)
2. (# 15 on p. 76 of Lang) Let G be a finite group acting transitively (i.e., there is only one orbit) on a set S, where S has at least 2 elements. Prove there is an element g in G with no fixed points (i.e., such that there is no s in S with gs = s).
3. (# 16 on p. 76 of Lang) Let H be a proper subgroup of a finite group G. Show that G is not the union of the conjugates of H.
4. Show that Problem 3 implies Problem 4 and vice versa (i.e., problems 3 and 4 are really the same problem).