Character triples and Shoda pairs
In this paper, a construction of Shoda pairs using character triples is given for a large class of monomial groups including abelian-by-supersolvable and subnormally monomial groups. The computation of primitive central idempotents and the structure of simple components of the rational group algebra for groups in this class are also discussed. The theory is illustrated with examples.
Keywords : rational group algebra, primitive central
idempotents, simple components, Shoda pairs, strong Shoda pairs, character triples, monomial groups.
MSC2000 : 16S34, 16K20, 16S35
Given a finite group , Shoda (, Corollary 45.4) gave a criterion to determine whether an induced monomial representation of is irreducible or not. Olivieri, del Río and Simón  rephrased Shoda’s theorem as follows:
If is a linear character of a subgroup of with kernel , then the induced character is irreducible if, and only if, the following hold:
(i) , is cyclic;
(ii) if and , then .
A pair of subgroups of satisfying (i) and (ii) above is called a Shoda pair of . For , define:
where runs over all the minimal normal subgroups of containing properly, and
An important feature (, Theorem 2.1) of a Shoda pair of is that there is a rational number , necessarily unique, such that is a primitive central idempotent of the rational group algebra , called the primitive central idempotent of realized by the Shoda pair . We’ll denote this by . For monomial groups, all the primitive central idempotents of are realized by Shoda pairs of . For the Shoda pair of , the case when is a primitive central idempotent of , is of special interest, thus, leading to the following definition of a strong Shoda pair. A strong Shoda pair  of is a pair of subgroups of satisfying the following conditions:
(ii) is cyclic and a maximal abelian subgroup of ;
(iii) the distinct -conjugates of are mutually orthogonal.
In , it is proved that if is a strong Shoda pair of , then it is also a Shoda pair of and is a primitive central idempotent of The groups such that all the primitive central idempotents of are realized by strong Shoda pairs of are termed as strongly monomial groups. Examples of such groups include abelian-by-supersolvable groups (, Theorem 4.4). The main reason for defining strong Shoda pairs in  was that the authors were able to provide a description of the structure of the simple component of for a strong Shoda pair of .
The work in  thus leads to the problem of computing Shoda pairs of a given finite group and to provide a description of the structure of the simple components of corresponding to the primitive central idempotents realized by them. The interest is in fact in providing a method to obtain a set of Shoda pairs of such that the mapping defines a bijection from to the set of all primitive central idempotents of realized by Shoda pairs of . Such a set is called a complete and irredundant set of Shoda pairs of , and has recently been provided by the first author with Maheshwary  for normally monomial groups. For the work in this direction, also see  and .
In this paper, we plan to study the problem for the class of all finite groups such that all the subgroups and quotient groups of satisfy the following property: either it is abelian or it contains a non central abelian normal subgroup. The groups in are known to be monomial (, Lemma 24.2). However, we have noticed that is not contained in the class of strongly monomial groups. Huppert (, Theorem 24.3) proved that contains all the groups for which there is a solvable normal subgroup of such that all Sylow subgroups of are abelian and is supersolvable. In particular, contains abelian-by-supersolvable groups. In view of an important criterion of subnormally monomial groups given in  and , we have shown in section 2 that also contains all subnormally monomial groups and, in particular, normally monomial groups. Our aim is to extend the work to the class .
An important tool which has turned out to be useful is Isaacs’s notion of character triples together with Clifford’s correspondence theorem. Following Isaacs (, p.186), we have defined -linear character triples of for a normal subgroup of . In view of Clifford’s correspondence theorem (, Theorem 6.11), for each -linear character triple of , we have defined its direct Clifford correspondents, which is another set of -linear character triples of with useful properties proved in Theorem 1 of section 3. With its help, we have given, in section 4, a construction of Shoda pairs of groups in . For each normal subgroup of , we have constructed a rooted directed tree , whose particular leaves correspond to Shoda pairs of , if (Theorem 2). We have also explored the condition for the collection of Shoda pairs corresponding to these leaves of as runs over all the normal subgroups of to be complete and irredundant. In section 5, we have given a new character free expression of , where is a Shoda pair of corresponding to a leaf of . This expression is in terms of the directed path from the root to the corresponding leaf and enables us to provide a necessary and sufficient condition for to be a primitive central idempotent of . In section 6, we generalize Proposition 3.4 of  and determine the structure of the simple components of for . Finally, in section 7, we provide illustrative examples.
2 The class of monomial groups
Throughout this paper, denotes a finite group. By , , , we mean, respectively, that is a subgroup, proper subgroup, normal subgroup of . Denote by , the index of in . Also denotes the normalizer of in and is the largest normal subgroup of contained in . For , is the commutator of and , and is the centralizer of in . Denote by , the set of all complex irreducible characters of . For a character of , and denotes the field obtained by adjoining to the character values , . If is a character of a subgroup of and , then is the character of given by , . Denote by , the character induced to . For a subgroup of , denotes the restriction of to .
Let denote the class of all finite groups such that all the subgroups and quotient groups of satisfy the following property: either it is abelian or it contains a non central abelian normal subgroup. It follows from Lemma 24.2 of  that the groups in are monomial. Recall that a finite group is monomial if every complex irreducible character of the group is induced by a linear character of a subgroup. In this section, we compare with the following classes of groups:
We prove the following:
The following statements hold:
Proof. (i) Clearly, Ab-by-Nil Ab-by-Sup -by-Sup. From (, Theorem 24.3), it follows that -by-Sup . This proves (i).
(ii) It is obvious that nM sM. We now show that Ab-by-Nil sM. Let Ab-by-Nil. Let be a normal abelian subgroup of such that is nilpotent. Let . It is already known that is monomial. By (, Lemma 24.8), there exists a subgroup of containing such that is induced from a linear character on . As is a subgroup of the nilpotent group , it is subnormal in . Consequently, is subnormal in . This proves that sM. Next, by (, Theorem 3.7), we have sM . We now show that . By Lemma 2.6 of , is closed under taking subgroups and factor groups. Thus to prove that , we only need to show that every non abelian group in contains a non central abelian normal subgroup. Let If is nilpotent, then clearly it has the desired property. If is not nilpotent, then Lemma 2.7 of  implies that , the socle of , is non central in . Also, in view of (, Lemma 3.11, Problem 2A.5), is abelian, as is solvable. Hence has the desired property and it follows that .
(iii) Consider the group generated by with defining relations: It is easy to see that is supersolvable and hence belongs to bySup. We’ll show that . Let be the class of all chiefly sub-Frobenius groups, i.e., all finite solvable groups for which is subnormal in whenever is an element of a chief factor of . It is easy to see that supersolvable groups are chiefly sub-Frobenius, and hence . Also it is known (, Theorem 3.8) that sM=. Now if , then it follows that is subnormally monomial. It can be shown that has an irreducible character of degree , denoted by say. If is subnormally monomial, then is induced from a linear character of a subnormal subgroup of . Also, implies that , which yields that is in fact normal in . However, does not have any normal subgroup of index , a contradiction. This proves that and (iii) follows.
(iv) Consider =SmallGroup in GAP library. It can be checked using GAP that is subnormally monomial and hence belongs to the class but it does not belong to -by-Sup.
(v) Simple computations using Wedderga reveal that SmallGroup is not strongly monomial. However, it belongs to . This proves (v).
3 -linear character triples
Let be a finite group. Let and a character triple, i.e., and is invariant in . We call it to be -linear character triple of , if is linear and . For -linear character triple of , denote by , the set of all irreducible characters of which lie above , i.e., the restriction of to has as a constituent. Let be its subset consisting of those which satisfy Denote by , the subset of consisting of linear characters. Further, for the character triple , we fix a normal subgroup of of maximal order containing such that is abelian. Note that there may be several choices of such , however, we fix one such choice for a given triple . Observe that always contains the center of . However, it can be seen that if and is non abelian, then properly contains the center of . We shall later use this observation without any mention.
Given - linear character triple of , we provide a construction of the set of another -linear character triples of required for the purpose of constructing Shoda pairs of .
Let be the group of automorphisms of the field of complex numbers which keep fixed. For brevity, denote by . Consider the action of on by setting
Also acts on by
Notice that the two actions on are compatible in the sense that
This consequently gives an action of on Under this double action, denote by , a set of representatives of distinct orbits of . If , set
where is the inertia group of in . For , define to be an empty set. Note that all the character triples in are -linear character triples of and we call them the direct Clifford correspondents (abbreviated d.c.c.) of . The name ‘direct Clifford correspondents’ refers to the fact that the characters in are Clifford correspondents of the characters in in view of the following theorem:
Let and a normal subgroup of . Let be -linear character triple of with and be as defined above. Let . Then
for any the following hold:
. Furthermore, holds, if, and only if, is abelian;
the induction defines an injective map from to .
for each , there exists , and such that
if , and are such that , then, and in this case for some .
Proof. (i) We first show that
If is abelian, then clearly and therefore the above equation holds trivially, as . If is non abelian, then properly contains the centre of . However, being invariant in , it follows that is contained in the centre of . Therefore, eqn (1) follows. This proves (a). Next, consider and Then
if, and only if,
However, if , then , as . This gives , as . Consequently, being in , eqn (2) follows. This proves (b). In view of Clifford’s correspondence theorem (, Theorem 6.11), defines an injective map from to . It can be checked that, under this map, is mapped to . This finishes the proof of (c).
(ii) Consider . Let be an irreducible constituent of We claim that . As and is invariant in , by (, Theorem 6.2), is the only irreducible constituent of . However, being an irreducible constituent of , it follows that is a constituent of . Hence is an irreducible constituent of , and therefore . But , as is normal in . This gives . Consequently, being abelian, it follows that is linear and moreover . We now show that . As , the inner product of with is non zero, and is irreducible, we have . Hence , which gives . Also, implies that . Hence the claim follows. Now choose which lies in the orbit of . This gives and such that . As is an irreducible constituent of it follows that is an irreducible constituent of the restriction of to . However, being normal in , from Clifford’s theorem (, Theorem 6.2), it follows that is an irreducible constituent of the restriction of to . Consequently, Clifford’s correspondence theorem provides such that It is easy to check that this belongs to . Hence (ii) follows.
(iii) Suppose are such that
where . By restricting to , it follows from (, Theorem 6.2) that
for some . Therefore, and lie in the same orbit under the double action and hence . In this case, from eqn (4),
which on comparing the kernels yields Now using eqn (5) and the fact that , it is easy to see that Thus and both belong to and, in view of eqn (3), they are same when induced to . Consequently, the injectivity of the induction map in part (i) implies . This proves (iii) and completes the proof.
4 A construction of Shoda pairs
We begin by recalling some basic terminology in graph theory. A graph is a pair , where is a non-empty set whose elements are termed vertices of and is a set of unordered pairs of vertices of . Each element , where , is called an edge and is said to join the vertices and . If , then is said to be incident with both and . Further, if , the degree of is the number of edges in that are incident with . A walk in the graph is a sequence of vertices and edges of the form : , where each edge is incident with the vertices and immediately preceding and succeeding it. A walk is termed path if all the vertices are distinct and is called a cycle if it begins and ends with the same vertex and all other vertices are distinct. A connected graph is the one in which any two vertices are connected by a path. A connected graph which contain no cycles is called a tree.
A directed graph is a pair , where is a non-empty set whose elements are termed vertices and is a set of ordered pairs of vertices of . In a directed graph, an edge is said to be incident out of and incident into . The terminology of directed walk and directed path is same as that in graph but now the edges are directed in the same direction. In a directed graph, the number of edges incident out of is called out-degree of and the number of edges incident into is called in-degree of , where . The vertices of in-degree are called source and those of out-degree are called sink. In a directed graph, there is an obvious underlying undirected graph whose vertex set is same as that of the directed graph and there is an edge if either or is an edge in the directed graph. A directed graph is called a directed tree if its underlying undirected graph is a tree. The sink vertices of a directed tree are termed as its leaves. A directed tree is called a rooted directed tree if it has a unique source. The unique source of a rooted directed tree is called its root.
We now proceed with the construction of Shoda pairs. Let and let be a normal subgroup of . Consider the directed graph whose vertex set consist of all -linear character triples of and there is an edge if is a direct Clifford correspondent(d.c.c.) of . Clearly , where is the character of which takes constant value . Let be the set of those vertices for which there is a directed path from to . Let be the set of ordered pairs with . Then is a directed subgraph of . Observe that any vertex of with is a sink vertex.
Let and the set of all normal subgroups of .
For , the following hold:
is a rooted directed tree with as its root;
the leaves of of the type correspond to Shoda pairs of . More precisely, if is a leaf of , then is a Shoda pair of .
If is any Shoda pair of , then there is a leaf of , where , such that and realize the same primitive central idempotent of .
To prove the theorem, we need some preparation.
For each vertex of , there is a unique directed path from to .
Proof. Let and be two directed paths from to . Assume that . We claim that . We’ll prove it by induction on . For , we already have . Assume that for some . Write and for . From the construction of d.c.c., we have and . As , it follows immediately that . Now (resp. ) being the restriction of to (resp. ) yields that . Further, as and , it follows that . This proves the claim, which as a consequence implies that both and are equal to . This is not possible if as no two vertices in a path are same.
The following statements hold for :
If and are edges of with , then ;
If is a path in the underlying undirected graph of , then there is a unique , , with the following:
Proof. (i) is a consequence of Lemma 1 and (ii) follows immediately from (i). It may be mentioned that in the lemma, (a) is empty if , and (b) is empty if .
The underlying undirected graph of is a tree.
Proof. Let be the underlying undirected graph of Clearly is a connected graph. To show that is a tree, it is enough to prove that is disconnected after removing an edge (see , Theorem 3-5). Let and let . We need to show that is disconnected. If not, then there is a path
in , where and . As , either or belongs to . Suppose . As the path given in eqn (6) is also a path in , there is a , so that part (ii) of Lemma 2 holds. If , then is a d.c.c. of , which by Lemma 2(i) implies . Consequently which is not so. If , then is a d.c.c. of for all . Hence if , then from Theorem 1, . However implies , a contradiction. Using similar arguments, it follows that is not possible. This proves the lemma.
We are now ready to prove the theorem. Recall from (, Proposition 1.1) that if , then is a primitive central idempotent of , where is the Galois group of over .
Proof of Theorem 2 (i) By Lemma 3, is a directed tree. If the in-degree of is non zero then there is a -linear character triple of such that is a d.c.c. of . Hence by Theorem 1(i)(a) . However being -linear character triple of , we have , which gives , a contradiction. This proves that is a source of . Next if is a vertex of different from , then there is a directed path from to , which implies that the in-degree of is non zero. Hence can’t be a source. This proves (a). To prove (b), consider a leaf of of the type . Let be the directed path from to . Let , . As is a d.c.c. of , from Theorem 1
This is true for all . Observe that , as . The repeated application of eqn (7) when is implies that
For , the above equation, in particular, gives . This proves (b).
(ii) Let be a Shoda pair of and let