- group, ∞-group
- group object, group object in an (∞,1)-category
- abelian group, spectrum
- super abelian group
- group action, ∞-action
- representation, ∞-representation
- progroup
- homogeneous space

**Classical groups**

**Finite groups**

**Group schemes**

**Topological groups**

**Lie groups**

**Super-Lie groups**

**Higher groups**

**Cohomology and Extensions**

**Related concepts**

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

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

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.

- for the primary components of the stable homotopy groups of spheres see at
*homotopy groups of spheres – Table*.

- John Sullivan,
*Classification of finite abelian groups*(pdf)

Last revised on December 8, 2022 at 22:59:16. See the history of this page for a list of all contributions to it.