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The degree of some algebraic structure is some parameter that describes the structure. There is no universal definition valid for all structures, it is a per structure type thing.
This is particularly useful when talking about structures with an infinite number of elements, but it is sometimes also used for finite structures.
Examples:
- the dihedral group of degree n acts on n elements, and has order 2n
- the parameter that characterizes the size of the general linear group is called the degree of that group, i.e. the dimension of the underlying matrices