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A cyclic group is a group which is equal to one of its cyclic subgroups: G = g for some element g, called a generator of G. For a finite cyclic group G of order n we have G = {e, g, g 2, ... , g n−1}, where e is the identity element and g i = g j whenever i ≡ j (mod n); in particular g n = g 0 = e, and g −1 = g n−1.
In mathematics, the classification of finite simple groups (popularly called the enormous theorem [1] [2]) is a result of group theory stating that every finite simple group is either cyclic, or alternating, or belongs to a broad infinite class called the groups of Lie type, or else it is one of twenty-six exceptions, called sporadic (the Tits group is sometimes regarded as a sporadic group ...
For every finite group G of order n, the following statements are equivalent: . G is cyclic.; For every divisor d of n, G has at most one subgroup of order d.; If either (and thus both) are true, it follows that there exists exactly one subgroup of order d, for any divisor of n.
One can check that the cosets form a group of three elements (the product of a red element with a blue element is blue, the inverse of a blue element is green, etc.). Thus, the quotient group / is the group of three colors, which turns out to be the cyclic group with three elements.
cyclic group A cyclic group is a group that is generated by a single element, that is, a group such that there is an element g in the group such that every other element of the group may be obtained by repeatedly applying the group operation to g or its inverse.
Informally, G has the above presentation if it is the "freest group" generated by S subject only to the relations R. Formally, the group G is said to have the above presentation if it is isomorphic to the quotient of a free group on S by the normal subgroup generated by the relations R. As a simple example, the cyclic group of order n has the ...
p-groups of the same order are not necessarily isomorphic; for example, the cyclic group C 4 and the Klein four-group V 4 are both 2-groups of order 4, but they are not isomorphic. Nor need a p-group be abelian; the dihedral group Dih 4 of order 8 is a non-abelian 2-group. However, every group of order p 2 is abelian.
For example, in the symmetric group shown above, where ord(S 3) = 6, the possible orders of the elements are 1, 2, 3 or 6. The following partial converse is true for finite groups: if d divides the order of a group G and d is a prime number, then there exists an element of order d in G (this is sometimes called Cauchy's theorem).