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group theory

Contents

Definition

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

(Also called primary group a $p$-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 $p$-primary subgroups.

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

Nilpotency

Proposition

Every finite $p$-primary group $G$ has a nontrivial center $Z(G)$.

For a proof, see class equation.

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

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

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

Examples

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