Construction of Cayley table My tutor has asked me to describe the symmetries of non square rectangle and to construct Cayley table for it. Is it same as the klein 4 group?  
 A: You need a little care here as your question says "the" Klein 4 group (the word "the" means there is only one such object). Some people do use this wording to mean the specific subgroup of 4 permutations from $S_{4}$ given by $\{ (1), (12)(34), (13)(24), (14)(23) \}$ (which is commonly denoted by the letter $V$). In this case since symmetries of the rectangle (which is not a square) are NOT permutations in $S_{4}$ your group is not equal to "the" Klein 4 group, however it is isomorphic to it. (So if you could only see how elements of each these groups interacted with each other but not what the elements were labelled, you could would not be able to tell if you were looking at your group or $V$).
Some people only talk about "a" Klein 4 group, in which case they mean a "family" of groups, the members of which are  any group with 4 elements and is NOT cyclic. This is the same as saying any group isomorphic to $V$. So in this case the group of symmetries of the rectangle (which is not a square) is "a" Klein 4 group. With this notion some people use $\mathbb{K_{4}}$ to denote a "general" Klein 4 group (in much the same way that people tend to denote a "general" cyclic group with $n$ elements by $C_{n}$ whereas $\mathbb{Z}_{n}$ is a particular cyclic group with  $n$ elements).
