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For a prime number, a group is -primary if each of its elements has a prime power order .
(Also called primary group a -group, but NO relation to n-group.)
The fundamental theorem of finite abelian groups stats that every finite abelian group is a direct sum of its -primary subgroups.
These are often called its -primary parts or -primary components. See also at Adams spectral sequence and for instance at stable homotopy groups of spheres.
For a proof, see class equation.
Since the center is a normal subgroup of , we may define by induction (with the help of this proposition here) a series of inclusions of normal subgroups where is the trivial subgroup and is the inverse image of the center along the canonical homomorphism . The resulting series
is called the upper central series of , and Proposition shows that in the case of a finite -group, this series consists of strict inclusions that eventually terminate in the full subgroup . A group with that property is a nilpotent group. In particular it is a solvable group.
Last revised on December 8, 2022 at 22:59:16. See the history of this page for a list of all contributions to it.