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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).
Each group is named by Small Groups library as G o i, where o is the order of the group, and i is the index used to label the group within that order. Common group names: Z n: the cyclic group of order n (the notation C n is also used; it is isomorphic to the additive group of Z/nZ) Dih n: the dihedral group of order 2n (often the notation D n ...
In mathematics this group is known as the dihedral group of order 8, and is either denoted Dih 4, D 4 or D 8, depending on the convention. This was an example of a non-abelian group: the operation ∘ here is not commutative , which can be seen from the table; the table is not symmetrical about the main diagonal.
A more complex example involves the order of the smallest simple group that is not cyclic. Burnside's p a q b theorem states that if the order of a group is the product of one or two prime powers, then it is solvable, and so the group is not simple, or is of prime order and is cyclic. This rules out every group up to order 30 (= 2 · 3 · 5).
The theory of finite groups is the set of first-order statements in the language of groups that are true in all finite groups (there are plenty of infinite models of this theory). It is not completely trivial to find any such statement that is not true for all groups: one example is "given two elements of order 2, either they are conjugate or ...
The trivial group is the only group of order one, and the cyclic group C p is the only group of order p. There are exactly two groups of order p 2, both abelian, namely C p 2 and C p × C p. For example, the cyclic group C 4 and the Klein four-group V 4 which is C 2 × C 2 are both 2-groups of order 4.
Applications of group theory abound. Almost all structures in abstract algebra are special cases of groups. Rings, for example, can be viewed as abelian groups (corresponding to addition) together with a second operation (corresponding to multiplication). Therefore, group theoretic arguments underlie large parts of the theory of those entities.
It is the smallest finite non-abelian group. A common example from physics is the rotation group SO(3) in three dimensions (for example, rotating something 90 degrees along one axis and then 90 degrees along a different axis is not the same as doing them in reverse order). Both discrete groups and continuous groups may be non-abelian