Conjugate elements and conjugate subgroups

Wondratschek, H.

doi:10.1107/97809553602060000538

International
Tables for
Crystallography
Volume A1
Symmetry relations between space groups
Edited by Hans Wondratschek and Ulrich Müller

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International Tables for Crystallography (2006). Vol. A1. ch. 1.2, pp. 11-12 | 1 | 2 |

Section 1.2.4.3. Conjugate elements and conjugate subgroups

Hans Wondratschek^a ^*

^a Institut für Kristallographie, Universität, D-76128 Karlsruhe, Germany
Correspondence e-mail: wondra@physik.uni-karlsruhe.de

1.2.4.3. Conjugate elements and conjugate subgroups

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In a coset decomposition, the set of all elements of the group $[{\cal G}]$ is partitioned into cosets which form classes in the mathematical sense of the word, i.e. each element of $[{\cal G}]$ belongs to exactly one coset.

Another equally important partition of the group $[{\cal G}]$ into classes of elements arises from the following definition:

Definition 1.2.4.3.1. Two elements $[{\sf g}_j,{\sf g}_k\in{\cal G}]$ are called conjugate if there is an element $[{\sf g}_q\in{\cal G}]$ such that $[{\sf g}_q^{-1}{\sf g}_j\,{\sf g}_q={\sf g}_k]$ .

Remarks :

(1) Definition 1.2.4.3.1 partitions the elements of $[{\cal G}]$ into classes of conjugate elements which are called conjugacy classes of elements.
(2) The unit element always forms a conjugacy class by itself.
(3) Each element of an Abelian group forms a conjugacy class by itself.
(4) Elements of the same conjugacy class have the same order.
(5) Different conjugacy classes may contain different numbers of elements, i.e. have different `lengths'.

Not only the individual elements of a group $[{\cal G}]$ but also the subgroups of $[{\cal G}]$ can be classified in conjugacy classes.

Definition 1.2.4.3.2. Two subgroups $[{\cal H}_j,{\cal H}_k \,\lt\, {\cal G}]$ are called conjugate if there is an element $[{\sf g}_q\in{\cal G}]$ such that $[{\sf g}_q^{-1}{\cal H}_j\,{\sf g}_q={\cal H}_k]$ holds. This relation is often written $[{\cal H}_j^{{\sf g}_q}={\cal H}_k]$ .

Remarks :

(1) The `trivial subgroup' $[{\cal I}]$ (consisting only of the unit element of $[{\cal G}]$ ) and the group $[{\cal G}]$ itself each form a conjugacy class by themselves.
(2) Each subgroup of an Abelian group forms a conjugacy class by itself.
(3) Subgroups in the same conjugacy class are isomorphic and thus have the same order.
(4) Different conjugacy classes of subgroups may contain different numbers of subgroups, i.e. have different lengths.

Equation (1.2.4.1) can be written $[{\cal H}={\sf g}_p^{-1}{\cal H}\,{\sf g}_p \,\, {\rm or }\,\,{\cal H}={\cal H}^{{\sf g}_p}\,\,{ \rm for \,\,all }\,\, p;\ 1\le p\le i. \eqno (1.2.4.2)]$ Using conjugation, Definition 1.2.4.2.3 can be formulated as

Definition 1.2.4.3.3. A subgroup $[{\cal H}]$ of a group $[{\cal G}]$ is a normal subgroup $[{\cal H}\triangleleft{\cal G}]$ if it is identical with all of its conjugates, i.e. if its conjugacy class consists of the one subgroup $[{\cal H}]$ only.

References

International Tables for Crystallography (2006). Vol. A1. ch. 1.2, pp. 11-12