Coset decomposition and normal 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, p. 11 | 1 | 2 |

Section 1.2.4.2. Coset decomposition and normal subgroups

Hans Wondratschek^a ^*

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

1.2.4.2. Coset decomposition and normal subgroups

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Let $[{\cal H} \,\lt\, {\cal G}]$ be a subgroup of $[{\cal G}]$ of order $[|{\cal H}|]$ . Because $[{\cal H}]$ is a proper subgroup of $[{\cal G}]$ there must be elements $[{\sf g}_q\in{\cal G}]$ that are not elements of $[{\cal H}]$ . Let $[{\sf g}_2\in{\cal G}]$ be one of them. Then the set of elements $[{\sf g}_2\,{\cal H}=\{{\sf g}_2\,{\sf h}_j\,|\, {\sf h}_j\in{\cal H}\}]$ ³ is a subset of elements of $[{\cal G}]$ with the property that all its elements are different and that the sets $[{\cal H}]$ and $[{\sf g}_2\,{\cal H}]$ have no element in common. Thus, the set $[{\sf g}_2\,{\cal H}]$ also contains $[|{\cal H}|]$ elements of $[{\cal G}]$ . If there is another element $[{\sf g}_3\in{\cal G}]$ which belongs neither to $[{\cal H}]$ nor to $[{\sf g}_2\,{\cal H}]$ , one can form another set $[{\sf g}_3{\cal H}=\{{\sf g}_3{\sf h}_j\,|\,{\sf h}_j\in{\cal H}\}]$ . All elements of $[{\sf g}_3{\cal H}]$ are different and none occurs already in $[{\cal H}]$ or in $[{\sf g}_2\,{\cal H}]$ . This procedure can be continued until each element $[{\sf g}_r\in{\cal G}]$ belongs to one of these sets. In this way the group $[{\cal G}]$ can be partitioned, such that each element $[{\sf g}\in{\cal G}]$ belongs to exactly one of these sets.

Definition 1.2.4.2.1. The partition just described is called a decomposition ( $[{\cal G}]$ : $[{\cal H}]$ ) into left cosets of the group $[{\cal G}]$ relative to the group $[{\cal H}]$ . The sets $[{\sf g}_p\,{\cal H},\ p=1,\ \ldots,\ i]$ are called left cosets, because the elements $[{\sf h}_j\in {\cal H}]$ are multiplied with the new elements from the left-hand side. The procedure is called a decomposition into right cosets $[{\cal H}{\sf g}_s]$ if the elements $[{\sf h}_j\in{\cal H}]$ are multiplied with the new elements $[{\sf g}_s]$ from the right-hand side. The elements $[{\sf g}_p]$ or $[{\sf g}_s]$ are called the coset representatives. The number of cosets is called the index $[i=|{\cal G}:{\cal H}|]$ of $[{\cal H}]$ in $[{\cal G}]$ .

Remarks :

(1) The group $[{\cal H}={\sf g}_1{\cal H}]$ with $[{\sf g}_1={\sf e}]$ is the first coset for both kinds of decomposition. It is the only coset which forms a group by itself.
(2) All cosets have the same length, i.e. the same number of elements, which is equal to $[|{\cal H}|]$ , the order of $[{\cal H}]$ .
(3) The index i is the same for both right and left decompositions. In IT A and in this volume, the index is frequently designated by the symbol .
(4) A coset does not depend on its representative element; starting from any of its elements will result in the same coset. The right cosets may be different from the left ones and the representatives of the right and left cosets may also differ.
(5) If the order $[|{\cal G}|]$ of $[{\cal G}]$ is infinite, then either the order $[|{\cal H}|]$ of $[{\cal H}]$ or the index $[i=|{\cal G}:{\cal H}|]$ of $[{\cal H}]$ in $[{\cal G}]$ or both are infinite.
(6) The coset decomposition of a space group $[{\cal G}]$ relative to its translation subgroup $[{\cal T}]$ ( $[{\cal G}]$ ) is fundamental in crystallography, cf. Section 1.2.5.4.

From its definition and from the properties of the coset decomposition mentioned above, one immediately obtains the fundamental theorem of Lagrange (for another formulation, see Chapter 1.5 ):

Lemma 1.2.4.2.2. Lagrange's theorem: Let $[{\cal G}]$ be a group of finite order $[|{\cal G}|]$ and $[{\cal H} \,\lt\, {\cal G}]$ a subgroup of $[{\cal G}]$ of order $[|{\cal H}|]$ . Then $[|{\cal H}|]$ is a divisor of $[|{\cal G}|]$ and the equation $[|{\cal H}|\times{}i=|{\cal G}|]$ holds where $[i=|{\cal G}:{\cal H}|]$ is the index of $[{\cal H}]$ in $[{\cal G}]$ .

A special situation exists when the left and right coset decompositions of $[{\cal G}]$ relative to $[{\cal H}]$ result in the partition of $[{\cal G}]$ into the same cosets: $[{\sf g}_p\,{\cal H}={\cal H}\,{\sf g}_p \,\,{ \rm for\,\, all }\,\,1\le p\le i. \eqno (1.2.4.1)]$ Subgroups $[{\cal H}]$ that fulfil equation (1.2.4.1) are called `normal subgroups' according to the following definition:

Definition 1.2.4.2.3. A subgroup $[{\cal H} \,\lt\, {\cal G}]$ is called a normal subgroup or invariant subgroup of $[{\cal G}]$ , $[{\cal H}\triangleleft{\cal G}]$ , if equation (1.2.4.1) is fulfilled.

The relation $[{\cal H}\triangleleft{\cal G}]$ always holds for $[|{\cal G}:{\cal H}|=2]$ , i.e. subgroups of index 2 are always normal subgroups. The subgroup $[{\cal H}]$ contains half of the elements of $[{\cal G}]$ , whereas the other half of the elements forms `the other' coset. This coset must then be the right as well as the left coset.

References

International Tables for Crystallography (2006). Vol. A1. ch. 1.2, p. 11