(1) Let G be a group of order 94 = 2 x 47. Let Z(G), as usual, denote the center of G. (a) For each of the following, either just say "YES", or prove that the answer is always "No". (i) Can |Z(G)|=...

Extracted text: (1) Let G be a group of order 94 = 2 x 47. Let Z(G), as usual, denote the center of G. (a) For each of the following, either just say "YES", or prove that the answer is always "No". (i) Can |Z(G)|= 10? (ii) Can |Z(G)I = 47? (iii) Can |Z(G)| = 2? (b) Must G have a subgroup of order 2? Why or why not? (c) If G is abelian, can G have more than 1 element of order 2? Why or why not? (d) Must G have a subgroup of order 47? Why or why not? For full credit, justify your answers carefully and completely, and use Lagrange's theorem if and when necessary,