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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 ∗.
For example, in S 5, one cyclic subgroup of order 5 is generated by (13254), whereas the largest cyclic subgroups of S 5 are generated by elements like (123)(45) that have one cycle of length 3 and another cycle of length 2. This rules out many groups as possible subgroups of symmetric groups of a given size.
The Sylow theorems are a powerful statement about the structure of groups in general, but are also powerful in applications of finite group theory. This is because they give a method for using the prime decomposition of the cardinality of a finite group to give statements about the structure of its subgroups: essentially, it gives a technique ...
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 ...
The Brauer–Suzuki theorem about groups with generalized quaternion Sylow 2-subgroups shows in particular that none of them are simple. 1960 Thompson proves that a group with a fixed-point-free automorphism of prime order is nilpotent. 1960 Feit, Marshall Hall, and Thompson show that all finite simple CN groups of odd order are cyclic. 1960
Algebraic groups. v. t. e. In mathematics, more specifically in the field of group theory, a solvable group or soluble group is a group that can be constructed from abelian groups using extensions. Equivalently, a solvable group is a group whose derived series terminates in the trivial subgroup.