In the dihedral group D4, determine the inverse of each of Ï, Ï and ÏÏ. Show that (pT1. 2. a.) In the Klein 4-group, show that every element is its own inverse. b.) Show that, if every element of the group G is its own inverse, then G is Abelian. 3. Determine the order of each of the indicated elements in each of the indicated groups. Justify your answer. Do not simply assert a number. a.) 2 Z3 b.) 4 Z10 4. Give at least two examples of nontrivial proper subgroups of each of the groups Zs, S3 and GL(2,Q). 5. Show that, in an Abelian group G, the set F consisting of all elements of G of finite order is a subgroup of G. (It is not necessarily true that the set of all elements of finite order of a nonabelian group H is a subgroup of H. Thus, clearly indicate where you use the assumption that G is Abelian.)
