The groups Z/6Z, S3, GL(2,2), and De (the symmetries of an equilateral triangle) are all groups of order 6. Which ones are isomorphic? If two of the groups are isomorphic, give the relabeling explicitly. If two of the groups are not isomorphic, then give a reason.

Prove that two cycles (a1,a2,...ak), (b1,b2,...bm) are disjoint cycles.

Show that the group of rotations in R³ of a 3-prism (that is, a prism with equilateral ends, as in the following figure) is isomorphic to D3.

Which of the following groups are isomorphic? i) Z/75Z ii) Z/3Z e Z/25Z iii) Z/15Z e Z/5Z iv) Z/3Z e Z/5Z OZ/5Z.

plz 2Q and with details.