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## GroupProperties and Characterization

We have studied a variety of examples of groups in the previous chapter. We will now study some properties shared by all groups. It will be proved that in a groupoid if an identity element exists, it is unique. In examples of groups it was observed that every element had only one inverse. This was not by chance. In fact, we shall prove that, in a group, every element has an unique inverse.

#### 6.1 Properties of Groups

Before discussing the properties of groups, some notations, which will be used throughout, will be in order.

Notation: For a group (G, *) it is tedious to keep on writing the operation * throughout our calculations. Thus, except where necessary, juxtaposition will be used for the binary ...

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