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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).
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.
Initial work pointed towards the affirmative answer. For example, if a group G is finitely generated and the order of each element of G is a divisor of 4, then G is finite. . Moreover, A. I. Kostrikin was able to prove in 1958 that among the finite groups with a given number of generators and a given prime exponent, there exists a largest o
Depending on the prime factorization of n, some restrictions may be placed on the structure of groups of order n, as a consequence, for example, of results such as the Sylow theorems. For example, every group of order pq is cyclic when q < p are primes with p − 1 not divisible by q. For a necessary and sufficient condition, see cyclic number ...
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 ...
For example, the dihedral group D 8 of order sixteen can be generated by a rotation, r, of order 8; and a flip, f, of order 2; and certainly any element of D 8 is a product of r ' s and f ' s. However, we have, for example, rfr = f −1, r 7 = r −1, etc., so such products are not unique in D 8. Each such product equivalence can be expressed ...
In a finite group, some non-zero power of a must be the group identity, which we denote either as e or 1; the lowest such power is the order of the element a, the number of distinct elements in the cycle that it generates. In a cycle graph, the cycle is represented as a polygon, with its vertices representing the group elements and its edges ...