# Subgroup

### From Online Dictionary of Crystallography

Sous-groupe (*Fr*). Untergruppe (*Ge*). Subgrupo (*Sp*). Sottogruppo (*It*). 部分群 (*Ja*).

Let *G* be a group and *H* a non-empty subset of *G*. Then *H* is called a **subgroup** of *G* if the elements of *H* obey the group postulates, *i.e.* if

- the identity element
*1*of_{G}*G*is contained in*H*; -
*H*is closed under the group operation (inherited from*G*); -
*H*is closed under taking inverses.

The subgroup *H* is called a **proper subgroup** of *G* if there are elements of *G* not contained in *H*.

A subgroup *H* of *G* is called a **maximal subgroup** of *G* if there is no proper subgroup *M* of *G* such that *H* is a proper subgroup of *M*.

## See also

- Complex
- Coset
- Normal subgroup
- Supergroup
- Chapter 1.7.1 of
*International Tables for Crystallography, Volume A*, 6th edition