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 ...
