Conjugate elements and conjugate subgroups
Wondratschek, H.,

International Tables for Crystallography
(2011).
Vol. A1,
Section 1.2.4.3,
p.

[ doi:10.1107/97809553602060000791 ]
numbers of subgroups, i.e. have different lengths.
Equation (

**1.2.4.1**) can be written Using conjugation, Definition

**1.2.4.2.3** can be formulated as
Definition

**1.2.4.3.3**. A subgroup of a group ...

Definition
Wondratschek, H.,

International Tables for Crystallography
(2011).
Vol. A1,
Section 1.2.4.1,
p.

[ doi:10.1107/97809553602060000791 ]
introduction to the subgroups of space groups
There may be sets of elements that do not constitute the full group but nevertheless fulfil the group postulates for themselves.
Definition

**1.2.4.1.1**. A subset of elements ...

Subgroups
Wondratschek, H.,

International Tables for Crystallography
(2011).
Vol. A1,
Section 1.2.4,
p.

[ doi:10.1107/97809553602060000791 ]
of into the same cosets: Subgroups that fulfil equation (

**1.2.4.1**) are called `normal subgroups' according to the following definition:
Definition

**1.2.4.2.3**. A subgroup is called a normal subgroup or invariant subgroup ...

Coset decomposition and normal subgroups
Wondratschek, H.,

International Tables for Crystallography
(2011).
Vol. A1,
Section 1.2.4.2,
p.

[ doi:10.1107/97809553602060000791 ]
subgroups' according to the following definition:
Definition

**1.2.4.2.3**. A subgroup is called a normal subgroup or invariant subgroup of,, if equation (

**1.2.4.1**) is fulfilled.
The relation always ...

Factor groups and homomorphism
Wondratschek, H.,

International Tables for Crystallography
(2011).
Vol. A1,
Section 1.2.4.4,
p.

[ doi:10.1107/97809553602060000791 ]
introduction to the subgroups of space groups
For the following definition, the `product of sets of group elements' will be used:
Definition

**1.2.4.4.1**. Let be a group and, be two arbitrary sets of its elements which ...

Normalizers
Wondratschek, H.,

International Tables for Crystallography
(2011).
Vol. A1,
Section 1.2.4.5,
p.

[ doi:10.1107/97809553602060000791 ]
Definition

**1.2.4.5.1**. The set of all elements that map the subgroup onto itself by conjugation,, forms a group, called the normalizer of in, where .
Remarks
(1) The group is a normal ...