<P> The dihedral group (discussed above) is a finite group of order 8 . The order of r is 4, as is the order of the subgroup R it generates (see above). The order of the reflection elements f etc. is 2 . Both orders divide 8, as predicted by Lagrange's theorem . The groups F above have order p − 1 . </P> <P> Mathematicians often strive for a complete classification (or list) of a mathematical notion . In the context of finite groups, this aim leads to difficult mathematics . According to Lagrange's theorem, finite groups of order p, a prime number, are necessarily cyclic (abelian) groups Z . Groups of order p can also be shown to be abelian, a statement which does not generalize to order p, as the non-abelian group D of order 8 = 2 above shows . Computer algebra systems can be used to list small groups, but there is no classification of all finite groups . An intermediate step is the classification of finite simple groups . A nontrivial group is called simple if its only normal subgroups are the trivial group and the group itself . The Jordan--Hölder theorem exhibits finite simple groups as the building blocks for all finite groups . Listing all finite simple groups was a major achievement in contemporary group theory . 1998 Fields Medal winner Richard Borcherds succeeded in proving the monstrous moonshine conjectures, a surprising and deep relation between the largest finite simple sporadic group--the "monster group"--and certain modular functions, a piece of classical complex analysis, and string theory, a theory supposed to unify the description of many physical phenomena . </P> <P> Many groups are simultaneously groups and examples of other mathematical structures . In the language of category theory, they are group objects in a category, meaning that they are objects (that is, examples of another mathematical structure) which come with transformations (called morphisms) that mimic the group axioms . For example, every group (as defined above) is also a set, so a group is a group object in the category of sets . </P> <P> Some topological spaces may be endowed with a group law . In order for the group law and the topology to interweave well, the group operations must be continuous functions, that is, g h, and g must not vary wildly if g and h vary only little . Such groups are called topological groups, and they are the group objects in the category of topological spaces . The most basic examples are the reals R under addition, (R ∖ (0),), and similarly with any other topological field such as the complex numbers or p - adic numbers . All of these groups are locally compact, so they have Haar measures and can be studied via harmonic analysis . The former offer an abstract formalism of invariant integrals . Invariance means, in the case of real numbers for example: </P>

When do we considered a group to be a set
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