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?

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Yes it is. There are only two essentially different groups of order 4. This one and the cyclic one.

– Max Freiburghaus

Oct 20 at 19:03

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1 Answer

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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 S4S_{4} given by {(1),(12)(34),(13)(24),(14)(23)}\{ (1), (12)(34), (13)(24), (14)(23) \} (which is commonly denoted by the letter VV). In this case since symmetries of the rectangle (which is not a square) are NOT permutations in S4S_{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 VV).

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 VV. 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 K4\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 nn elements by CnC_{n} whereas Zn\mathbb{Z}_{n} is a particular cyclic group with nn elements).