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There are many different areas of mathematics: geometry, number theory, topology, algebra - and the problems in these fields can be very different from one another. But in the late 1800s, people noticed they were using the same tool to solve many different kinds of problems. This sparked an idea: instead of reinventing the wheel, why not lift the tool out of the different branches of math and study the full power of the tool in a general form? This led to the creation of a new subject called Abstract Algebra, and the star of this field is a structure called the Group. Before we give the definition of a group, lets see a few examples. This will highlight the wide variety of groups and help we appreciate why it took a while for mathematicians to recognize that they were using the same tool in completely different settings. To begin, imagine you live on an alien planet where each day is seven hours long. Here, a clock has seven hours. You might be tempted to number them one through s