1. Statement 1 | Every permutation is a cycle. Statement 2 | Every cycle is a permutation.
2. Find all zeros in the indicated finite field of the given polynomial with coefficients in that field. x^3 + 2x + 2 in Z_7
3. Using Fermat's theorem, find the remainder of 3^47 when it is divided by 23.
4. Statement 1 | There exists a free abelian group of every positive integer rank. Statement 2 | A finitely generated abelian group is free abelian if its Betti number equals the number of elements in some generating set.
5. Statement 1 | A factor group of a non-Abelian group is non-Abelian. Statement 2 | If K is a normal subgroup of H and H is a normal subgroup of G, then K is a normal subgroup of G.
