Equivalence relation on a set, partition of a set
Janovec, V.,
Hahn, Th. and
Klapper, H.,
International Tables for Crystallography
(2013).
Vol. D,
Section 3.2.3.1.4,
p.
[ doi:10.1107/97809553602060000916 ]
it is easy to corroborate that the relation on the set of integers fulfils all three conditions (
3.2.3.3) to (
3.2.3.5) and is, therefore, an equivalence relation on the set . On the other hand, the relation is not an equivalence relation ...
Normalizers
Janovec, V.,
Hahn, Th. and
Klapper, H.,
International Tables for Crystallography
(2013).
Vol. D,
Section 3.2.3.2.5,
p.
[ doi:10.1107/97809553602060000916 ]
of,
The normalizer determines the subgroups conjugate to under G (see Example
3.2.3.10). The number m of subgroups conjugate to a subgroup under G equals the index of in G : where the last equation holds for finite G ...
Groups and subgroups
Janovec, V.,
Hahn, Th. and
Klapper, H.,
International Tables for Crystallography
(2013).
Vol. D,
Section 3.2.3.2,
p.
[ doi:10.1107/97809553602060000916 ]
the manual for GI KoBo -1).
3.2.3.2.2. Subgroups
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Definition
3.2.3.3. Let G be a group. A subset F of G is a subgroup of G if it forms a group under the product rule of G ...
Orbits
Janovec, V.,
Hahn, Th. and
Klapper, H.,
International Tables for Crystallography
(2013).
Vol. D,
Section 3.2.3.3.3,
p.
[ doi:10.1107/97809553602060000916 ]
of a subgroup in Example [aS]
3.2.3.11 is the set of all subgroups conjugate under G to,
From Proposition
3.2.3.13 and from Example [oS]
3.2.3.22, it follows that stabilizers of objects from one orbit constitute ...
Intermediate subgroups and partitions of an orbit into suborbits
Janovec, V.,
Hahn, Th. and
Klapper, H.,
International Tables for Crystallography
(2013).
Vol. D,
Section 3.2.3.3.5,
p.
[ doi:10.1107/97809553602060000916 ]
of the correspondence (
3.2.3.69) of Proposition
3.2.3.23 on the successive decomposition (
3.2.3.25) . Derivation of the second part of Proposition
3.2.3.30 can be sketched in the following way: where the relation (
3.2.3.70) is used.
We note ...
Group action
Janovec, V.,
Hahn, Th. and
Klapper, H.,
International Tables for Crystallography
(2013).
Vol. D,
Section 3.2.3.3.1,
p.
[ doi:10.1107/97809553602060000916 ]
.
We must note that the replacement of the explicit mapping (
3.2.3.46) by a contracted version (
3.2.3.47) is not always possible (see Example [aS]
3.2.3.11).
The condition (
3.2.3.49) requires that the first action followed ...
Orbits of ordered pairs and double cosets
Janovec, V.,
Hahn, Th. and
Klapper, H.,
International Tables for Crystallography
(2013).
Vol. D,
Section 3.2.3.3.6,
p.
[ doi:10.1107/97809553602060000916 ]
is defined by the following relation: The requirements (
3.2.3.47) to (
3.2.3.49) are fulfilled, mapping (
3.2.3.97) defines an action of group G on the set .
The group action (
3.2.3.97) introduces the G -equivalence of ordered ...
Normal subgroups
Janovec, V.,
Hahn, Th. and
Klapper, H.,
International Tables for Crystallography
(2013).
Vol. D,
Section 3.2.3.2.6,
p.
[ doi:10.1107/97809553602060000916 ]
...
Subgroups
Janovec, V.,
Hahn, Th. and
Klapper, H.,
International Tables for Crystallography
(2013).
Vol. D,
Section 3.2.3.2.2,
p.
[ doi:10.1107/97809553602060000916 ]
and domain structures
Definition
3.2.3.3. Let G be a group. A subset F of G is a subgroup of G if it forms a group under the product rule of G, i.e. if it fulfils the group postulates (1) to (4 ...
Action of a group on a set
Janovec, V.,
Hahn, Th. and
Klapper, H.,
International Tables for Crystallography
(2013).
Vol. D,
Section 3.2.3.3,
p.
[ doi:10.1107/97809553602060000916 ]
of the correspondence (
3.2.3.69) of Proposition
3.2.3.23 on the successive decomposition (
3.2.3.25) . Derivation of the second part of Proposition
3.2.3.30 can be sketched in the following way: where the relation (
3.2.3.70) is used.
We note ...
