nLab
p-primary group

Contents

Contents

Definition

For pp a prime number, a group is pp-primary if each of its elements gg has order a power p n(g)p^{n(g)} of the given prime.

(Also called primary group a pp-group, but NO relation to n-group.)

Properties

Relation to finite abelian group

The fundamental theorem of finite abelian groups stats that every finite abelian group is a direct sum of its pp-primary subgroups.

These are often called its pp-primary parts or pp-primary components. See also at Adams spectral sequence and for instance at stable homotopy groups of spheres.

Nilpotency

Proposition

Every finite pp-primary group GG has a nontrivial center Z(G)Z(G).

For a proof, see class equation.

Since the center Z(G)Z(G) is a normal subgroup of GG, we may define by induction (with the help of this proposition here) a series of inclusions of normal subgroups Z k(G)GZ^k(G) \subseteq G where Z 0(G)Z^0(G) is the trivial subgroup and Z k(G)Z^k(G) is the inverse image of the center Z(G/Z k1(G))Z(G/Z^{k-1}(G)) along the canonical homomorphism GG/Z k1(G)G \to G/Z^{k-1}(G). The resulting series

Z 0(G)Z k1(G)Z k(G)Z^0(G) \subseteq \ldots \subseteq Z^{k-1}(G) \subseteq Z^k(G) \subseteq \ldots

is called the upper central series of GG, and Proposition shows that in the case of a finite pp-group, this series consists of strict inclusions that eventually terminate in the full subgroup GG. A group with that property is a nilpotent group. In particular it is a solvable group.

Examples

References

  • John Sullivan, Classification of finite abelian groups (pdf)

Last revised on January 9, 2017 at 09:53:26. See the history of this page for a list of all contributions to it.