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The fundamental theorem of abelian groups states that every finitely generated abelian group is a finite direct product of primary cyclic and infinite cyclic groups. Because a cyclic group is abelian, each of its conjugacy classes consists of a single element. A cyclic group of order n therefore has n conjugacy classes.
Abelian group. In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is commutative. With addition as an operation, the integers and the real numbers form abelian ...
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. [note 1]
Stated differently the fundamental theorem says that a finitely generated abelian group is the direct sum of a free abelian group of finite rank and a finite abelian group, each of those being unique up to isomorphism. The finite abelian group is just the torsion subgroup of G. The rank of G is defined as the rank of the torsion-free part of G ...
Order p: The only group is cyclic. Order p 2: There are just two groups, both abelian. Order p 3: There are three abelian groups, and two non-abelian groups. One of the non-abelian groups is the semidirect product of a normal cyclic subgroup of order p 2 by a cyclic group of order p.
In mathematics, specifically group theory, Cauchy's theorem states that if G is a finite group and p is a prime number dividing the order of G (the number of elements in G), then G contains an element of order p. That is, there is x in G such that p is the smallest positive integer with xp = e, where e is the identity element of G.
Atlas of Finite Group Representations: contains representations and other data for many finite simple groups, including the sporadic groups. Orders of non abelian simple groups up to 10 10, and on to 10 48 with restrictions on rank.
It is the kernel of the signature group homomorphism sgn : S n → {1, −1} explained under symmetric group. The group A n is abelian if and only if n ≤ 3 and simple if and only if n = 3 or n ≥ 5. A 5 is the smallest non-abelian simple group, having order 60, and thus the smallest non-solvable group.