Search results
Results From The WOW.Com Content Network
Subgroups of cyclic groups. In abstract algebra, every subgroup of a cyclic group is cyclic. Moreover, for a finite cyclic group of order n, every subgroup's order is a divisor of n, and there is exactly one subgroup for each divisor. [1][2] This result has been called the fundamental theorem of cyclic groups. [3][4]
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, g2, ... , gn−1}, where e is the identity element and gi = gj whenever i ≡ j (mod n); in particular gn = g0 = e, and g−1 = gn−1.
Subgroup. In group theory, a branch of mathematics, a subset of a group G is a subgroup of G if the members of that subset form a group with respect to the group operation in G. Formally, given a group G under a binary operation ∗, a subset H of G is called a subgroup of G if H also forms a group under the operation ∗.
In the quaternion group of order 8, each of the cyclic subgroups of order 4 is normal, but none of these are characteristic. However, the subgroup, {1, −1}, is characteristic, since it is the only subgroup of order 2. If n is even, the dihedral group of order 2n has 3 subgroups of index 2, all of which are normal. One of these is the cyclic ...
Definitions. A group G is called boundedly generated if there exists a finite subset S of G and a positive integer m such that every element g of G can be represented as a product of at most m powers of the elements of S : where and are integers. The finite set S generates G, so a boundedly generated group is finitely generated .
A simple illustration of Sylow subgroups and the Sylow theorems are the dihedral group of the n-gon, D 2n. For n odd, 2 = 2 1 is the highest power of 2 dividing the order, and thus subgroups of order 2 are Sylow subgroups.
The fundamental theorem of finite abelian groups states that every finite abelian group can be expressed as the direct sum of cyclic subgroups of prime-power order; it is also known as the basis theorem for finite abelian groups. Moreover, automorphism groups of cyclic groups are examples of abelian groups. [13]
Another characterization is that a finite p-group in which there is a unique subgroup of order p is either cyclic or a 2-group isomorphic to generalized quaternion group. [13] In particular, for a finite field F with odd characteristic, the 2-Sylow subgroup of SL 2 ( F ) is non-abelian and has only one subgroup of order 2, so this 2-Sylow ...