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

International Tables for Crystallography (2006). Vol. A1, ch. 1.5, pp. 36-38   | 1 | 2 |

## Section 1.5.5. Maximal subgroups

Gabriele Nebea*

aAbteilung Reine Mathematik, Universität Ulm, D-89069 Ulm, Germany
Correspondence e-mail: nebe@mathematik.uni-ulm.de

### 1.5.5. Maximal subgroups

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#### 1.5.5.1. Maximal subgroups and primitive -sets

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To determine the maximal t-subgroups of a space group , essentially one has to calculate the maximal subgroups of the finite group . There are fast algorithms to calculate these maximal subgroups if this finite group is soluble (see Definition 1.5.5.2.1), which is the case for three-dimensional space groups. To explain this method and obtain theoretical consequences for the index of maximal subgroups in soluble space groups, we consider abstract groups again in this section.

For an arbitrary group , one has a fast method of checking whether a given subgroup of finite index is maximal by inspection of the -set of left cosets of in . Assume that and let with , and with , . Then the set may be written as Then permutes the lines of the rectangle above: For all and all , the left coset is equal to some for an . Hence the jth line is mapped onto the set

Definition 1.5.5.1.1. Let be a group and X a -set.

 (i) A congruence on X is a partition of X into non-empty subsets such that for all , , implies . (ii) The congruences and are called the trivial congruences. (iii) X is called a primitive -set if is transitive on X, and X has only the trivial congruences.

Hence the considerations above have proven the following lemma.

Lemma 1.5.5.1.2. Let be a subgroup of the group . Then is a maximal subgroup if and only if the -set is primitive.

The advantage of this point of view is that the groups having a faithful, primitive, finite -set have a special structure. It will turn out that this structure is very similar to the structure of space groups.

If X is a -set and is a normal subgroup of , then acts on the set of -orbits on X, hence is a congruence on X. If X is a primitive -set, then this congruence is trivial, hence or for all . This means that either acts trivially or transitively on X.

One obtains the following:

Theorem 1.5.5.1.3. [Theorem of Galois (ca 1830).] Let be a finite group and let X be a faithful, primitive -set. Assume that is an Abelian normal subgroup. Then

 (a) is a minimal normal subgroup of (i.e. for all , or ). (b) is an elementary Abelian p-group for some prime p and is a prime power. (c) and is the unique minimal normal subgroup of .

Proof. Let be an Abelian normal subgroup. Then acts faithfully and transitively on X. To establish a bijection between the sets and X, choose and define . Since is transitive, is surjective. To show the injectivity of , let with . Then , hence . But then acts trivially on X, because if then the transitivity of implies that there is an with . Then , since is Abelian. Since X is a faithful -set, this implies and therefore . This proves . Since this equality holds for all nontrivial Abelian normal subgroups of , statement (a) follows. If p is some prime dividing , then the Sylow p-subgroup of is normal in , since is Abelian. Therefore it is also a characteristic subgroup of and hence a normal subgroup in (see the remarks below Definition 1.5.3.5.3). Since is a minimal normal subgroup of , this implies that is equal to its Sylow p-subgroup. Therefore, the order of is a prime power for some prime p and . Similarly, the set is a normal subgroup of properly contained in . Therefore and is elementary Abelian. This establishes (b).

To see that (c) holds, let . Choose . Then . Since acts transitively, there is an such that . Hence . As above, let be any element of X. Then there is an element with . Hence . Since z was arbitrary and X is faithful, this implies that . Therefore . Since is Abelian, one has , hence . To see that is unique, let be another normal subgroup of . Since is a minimal normal subgroup, one has , and therefore for , : . Hence centralizes , , which is a contradiction. QED

Hence the groups that satisfy the hypotheses of the theorem of Galois are certain subgroups of an affine group over a finite field . This affine group is defined in a way similar to the affine group over the real numbers where one has to replace the real numbers by this finite field. Then is the translation subgroup of isomorphic to the n-dimensional vector space over . The set X is the corresponding affine space . The factor group is isomorphic to a subgroup of the linear group of that does not leave invariant any non-trivial subspace of .

#### 1.5.5.2. Soluble groups

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Definition 1.5.5.2.1. Let be a group. The derived series of is the series defined via , . The group is called the derived subgroup of . The group is called soluble if for some .

Remarks

 (i) The are characteristic subgroups of . (ii) is Abelian if and only if . (iii) is characterized as the smallest normal subgroup of , such that is Abelian, in the sense that every normal subgroup of with an Abelian factor group contains . (iv) Subgroups and factor groups of soluble groups are soluble. (v) If is a normal subgroup, then is soluble if and only if and are both soluble.

#### Example 1.5.5.2.2

The derived series of is: (or in Hermann–Mauguin notation ) and that of is (Hermann–Mauguin notation: ).

Hence these two groups are soluble. (For an explanation of the groups that occur here and later, see Section 1.5.3.6.)

Now let be a three-dimensional space group. Then is an Abelian normal subgroup, hence is soluble. The factor group is isomorphic to a subgroup of either or and therefore also soluble. Using the remark above, one deduces that all three-dimensional space groups are soluble.

Lemma 1.5.5.2.3. Let be a three-dimensional space group. Then is soluble.

#### 1.5.5.3. Maximal subgroups of soluble groups

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Now let be a soluble group and a maximal subgroup of finite index in . Then the set of left cosets is a primitive finite -set. Let be the kernel of the action of on X. Then the factor group acts faithfully on X. In particular, is a finite group and X is a primitive, faithful -set. Since is soluble, the factor group is also a soluble group. Let be the derived series of with . Then is an Abelian normal subgroup of . The theorem of Galois (Theorem 1.5.5.1.3) states that is an elementary Abelian p-group for some prime p and for some . Since , the order of X is the index of in . Therefore one gets the following theorem:

Theorem 1.5.5.3.1. If is a maximal subgroup of finite index in the soluble group , then its index is a prime power.

In the proof of Theorem 1.5.5.1.3, we have established a bijection between and the -set X, which is now . Taking the full pre-image of in , then one has and . Hence we have seen the first part of the following theorem:

Theorem 1.5.5.3.2. Let be a maximal subgroup of the soluble group . Then the factor group acts primitively and faithfully on , and there is a normal subgroup with and . Moreover, if is another subgroup of , with and , then is conjugate to .

#### Example 1.5.5.3.3

is the tetrahedral group from Section 1.5.3.2 and is the stabilizer of one of the four apices in the tetrahedron. Then and is a faithful -set which can be identified with the set of apices of the tetrahedron. The normal subgroup is the normal subgroup of Section 1.5.3.2.

Now let be as above, and take a Sylow 2-subgroup of . Then is the normal subgroup from Section 1.5.3.2 and .

These observations result in an algorithm for computing maximal subgroups of soluble groups :

 (1) compute normal subgroups [candidates for ]; (2) compute a minimal normal subgroup of ; (3) find as a complement of in .