Exam 1

1. Let G be a group and let g be one fixed element of G. Show that the map i_g, such that i_g(x)=gxg^-1 for x Î G, is an automorphism of G.

2. Describe all group isomorphisms f: Z₉® U₉ by giving the possible values for f(1).

3. Find all subgroups of the group Z₁₈, and draw the subgroup diagram for the subgroups.

4. Give a careful proof that the following property of a binary structure áS,*ñ is indeed a structural property: For each c Î S, the equation x*x=c has a solution x in S.

5. Draw a Cayley digraph of the group Z₁₀ using the generating set {4,5}.

6. Express the permutation

m = æ ç è 1 2 3 4 5 6 7 8 3 8 7 2 4 6 1 5 ö ÷ ø

in S₈ as a product of disjoint cycles, and then as a product of transpositions. Is m Î A₈?

7. Give a self-contained definition of a group.