The remaining six sporadic groups do not divide the order of the friendly giant, which are termed the pariahs ( Ly, O'N, Ru, J 4, J 3, and J 1). In the classification of finite simple groups, twenty of twenty-six sporadic groups in the happy family are part of three families of groups which divide the order of the friendly giant, the largest sporadic group: five first generation Mathieu groups, seven second generation subquotients of the Leech lattice, and eight third generation subgroups of the friendly giant. This category has a non-trivial functor to itself only for n = 6. This can also be expressed category theoretically: consider the category whose objects are the n element sets and whose arrows are the bijections between the sets. A closely related result is the following theorem: 6 is the only natural number n for which there is a construction of n isomorphic objects on an n-set A, invariant under all permutations of A, but not naturally in one-to-one correspondence with the elements of A. This automorphism allows us to construct a number of exceptional mathematical objects such as the S(5,6,12) Steiner system, the projective plane of order 4 and the Hoffman-Singleton graph. S 6, with 720 = 6 ! elements, is the only finite symmetric group which has an outer automorphism. The smallest non- abelian group is the symmetric group S 3 which has 3! = 6 elements. It is also the smallest Granville number, or S.
Since 6 equals the sum of its proper divisors, it is a perfect number 6 is the smallest of the perfect numbers. Six is the smallest positive integer which is neither a square number nor a prime number it is the second smallest composite number, behind 4 its proper divisors are 1, 2 and 3. It is a composite number and the smallest perfect number. 6 ( six) is the natural number following 5 and preceding 7.
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