Structure and representations of Q-groups.

*(English)*Zbl 0546.20005
Lecture Notes in Mathematics. 1084. Berlin etc.: Springer-Verlag. VI, 290 p. DM 38.50 (1984).

Let Q be the field of rational numbers. A Q-group is a finite group all of whose complex representations have rationally valued characters. All symmetric groups and all Weyl groups are Q-groups.

This monograph is the first broad exposition of questions related to structure and representations of Q-groups.

Contents: Chapter 1. General structural results. 1. Basic properties of Q-groups. 2. Structure of Q-groups having abelian or dihedral Sylow 2- subgroups. 3. Strong and involutary Q-groups. 4. Solvable Q-groups. 5. The partially ordered set defined by a Q-group.

Chapter 2. Constructions of Q-groups. 1. Wreath products. 2. Semi-direct products. 3. Application to the Weyl groups of types \(A_ n\), \(B_ n\) and \(D_ n\). 4. Theory of transversal permutation representations.

Chapter 3. Local characters. 1. Closed algebras. The local rings \(\Gamma (G)_ V\) and \(B(G)_{\Delta}\). 2. Local idempotents. 3. The combinatorics of p-classes. 4. Local restriction and local induction. 5. The local subgroup \(G_ V\). The local induction principle. 6. Local multiplicities.

Chapter 4. Rational representations of Q-groups. 1. The local invariants. 2. The local character ring \(\Gamma (G_ V)_ V\). 3. Local splitting.

Chapter 5. Application to Weyl groups of exceptional type. 1. \(F_ 4\). 2. \(E_ 6\). 3. \(E_ 7\). 4. \(E_ 8\). - In an appendix are given the conjugacy classes and character tables of exceptional Weyl groups.

If G is a Q-group and C is a cyclic subgroup of order n in G then \(| N_ G(C):C_ G(C)| =\phi (n)\) where \(\phi\) is Euler’s number- theoretical function. Let \(P\in Syl_ 2(G)\) be abelian (\(| G|\) is even for a Q-group G). Then exp P\(=2\). If p is an odd prime divisor of \(| G|\) then \(p\equiv 3(mod 4)\). So by Walter’s classification of non-solvable groups with abelian Sylow 2-subgroup we have: If a Sylow 2- subgroup P of a Q-group G is abelian then G is solvable. The author proves that in this case \(G=PG'\) and G’ is 3-group.

Let \(\tau\) be an involution in a Q-group G. \(\tau\) is called irreducible if \(\tau\) is the only involution in \(C_ G(\tau)\). Otherwise \(\tau\) is called reducible. G contains an irreducible involution iff a Sylow 2- subgroup of G is either \(Z_ 2\) or \(Q_ 8\). If \(Z_ 2\in Syl_ 2(G)\) and G contains an irreducible involution then G is a Frobenius group \(Z_ 2E_ 3\) where \(E_ 3\) is an elementary abelian 3-group. The author also gives a complete classification of Q-groups G with \(Q_ 8\in Syl_ 2(G) (Q_ 8\) is the quaternion group of order 8).

R. Gow proved the following remarkable result. A solvable Q-group is a \(\{\) 2,3,5\(\}\) -group. If all representations of G are realizable over Q and G is solvable then G is a \(\{\) 2,3\(\}\) -group. The author gives a detailed outline of this result.

In chapter 2 two constructions are discussed for obtaining new Q-groups: wreath products and semi-direct products. If \(G=A wr B\) is a Q-group then A and B are Q-groups. This results is somewhat surprising. Note that \(G=Z_ 2 wr S_ 3 (S_ 3\) in regular representation) is not a Q-group. The author gives conditions under which a wreath product is a Q-group. For example the natural wreath product \(S_ m wr S_ n\) is a Q-group.

Let V be a finite-dimensional vector space over \(k=GF(p^ t)\), \(G\leq GL(V)\) be a subgroup of GL(V). The author determines conditions under which GV (semi-direct product) is a Q-group. By this condition the Frobenius group \(Q_ 8(Z_ 3\times Z_ 3)\) is a Q-group. As application of this result it is proved that the classical Weyl groups and their Sylow 2-subgroups are Q-groups. It is proved that Sylow 2- subgroups of alternating groups \(A_ n\) are Q-groups.

In chapter 3 are developed ring theoretic methods of investigation of Q- groups.

Chapter 4 presents the central topics of this book. It is devoted to an investigation of the rationally represented characters of a Q-group and their relationship to the permutation characters of the group. An example of such investigation is Artin’s famous induction theorem (if \(\chi\) is a rationally represented character of a finite group G then \(| G|_{\chi}\) may be written as a \({\mathbb{Z}}\)-linear combination of permutation characters of G induced from cyclic subgroups). The most general result in this direction is the induction theorem due to S. D. Berman and E. Witt. Chapter 4 uses techniques of algebraic geometry. Chapter 5 is application to exceptional Weyl groups of the methods of the previous chapter.

This book is an interesting application of character theory to finite groups. It gives much information on the mysterious class of Q-groups.

This monograph is the first broad exposition of questions related to structure and representations of Q-groups.

Contents: Chapter 1. General structural results. 1. Basic properties of Q-groups. 2. Structure of Q-groups having abelian or dihedral Sylow 2- subgroups. 3. Strong and involutary Q-groups. 4. Solvable Q-groups. 5. The partially ordered set defined by a Q-group.

Chapter 2. Constructions of Q-groups. 1. Wreath products. 2. Semi-direct products. 3. Application to the Weyl groups of types \(A_ n\), \(B_ n\) and \(D_ n\). 4. Theory of transversal permutation representations.

Chapter 3. Local characters. 1. Closed algebras. The local rings \(\Gamma (G)_ V\) and \(B(G)_{\Delta}\). 2. Local idempotents. 3. The combinatorics of p-classes. 4. Local restriction and local induction. 5. The local subgroup \(G_ V\). The local induction principle. 6. Local multiplicities.

Chapter 4. Rational representations of Q-groups. 1. The local invariants. 2. The local character ring \(\Gamma (G_ V)_ V\). 3. Local splitting.

Chapter 5. Application to Weyl groups of exceptional type. 1. \(F_ 4\). 2. \(E_ 6\). 3. \(E_ 7\). 4. \(E_ 8\). - In an appendix are given the conjugacy classes and character tables of exceptional Weyl groups.

If G is a Q-group and C is a cyclic subgroup of order n in G then \(| N_ G(C):C_ G(C)| =\phi (n)\) where \(\phi\) is Euler’s number- theoretical function. Let \(P\in Syl_ 2(G)\) be abelian (\(| G|\) is even for a Q-group G). Then exp P\(=2\). If p is an odd prime divisor of \(| G|\) then \(p\equiv 3(mod 4)\). So by Walter’s classification of non-solvable groups with abelian Sylow 2-subgroup we have: If a Sylow 2- subgroup P of a Q-group G is abelian then G is solvable. The author proves that in this case \(G=PG'\) and G’ is 3-group.

Let \(\tau\) be an involution in a Q-group G. \(\tau\) is called irreducible if \(\tau\) is the only involution in \(C_ G(\tau)\). Otherwise \(\tau\) is called reducible. G contains an irreducible involution iff a Sylow 2- subgroup of G is either \(Z_ 2\) or \(Q_ 8\). If \(Z_ 2\in Syl_ 2(G)\) and G contains an irreducible involution then G is a Frobenius group \(Z_ 2E_ 3\) where \(E_ 3\) is an elementary abelian 3-group. The author also gives a complete classification of Q-groups G with \(Q_ 8\in Syl_ 2(G) (Q_ 8\) is the quaternion group of order 8).

R. Gow proved the following remarkable result. A solvable Q-group is a \(\{\) 2,3,5\(\}\) -group. If all representations of G are realizable over Q and G is solvable then G is a \(\{\) 2,3\(\}\) -group. The author gives a detailed outline of this result.

In chapter 2 two constructions are discussed for obtaining new Q-groups: wreath products and semi-direct products. If \(G=A wr B\) is a Q-group then A and B are Q-groups. This results is somewhat surprising. Note that \(G=Z_ 2 wr S_ 3 (S_ 3\) in regular representation) is not a Q-group. The author gives conditions under which a wreath product is a Q-group. For example the natural wreath product \(S_ m wr S_ n\) is a Q-group.

Let V be a finite-dimensional vector space over \(k=GF(p^ t)\), \(G\leq GL(V)\) be a subgroup of GL(V). The author determines conditions under which GV (semi-direct product) is a Q-group. By this condition the Frobenius group \(Q_ 8(Z_ 3\times Z_ 3)\) is a Q-group. As application of this result it is proved that the classical Weyl groups and their Sylow 2-subgroups are Q-groups. It is proved that Sylow 2- subgroups of alternating groups \(A_ n\) are Q-groups.

In chapter 3 are developed ring theoretic methods of investigation of Q- groups.

Chapter 4 presents the central topics of this book. It is devoted to an investigation of the rationally represented characters of a Q-group and their relationship to the permutation characters of the group. An example of such investigation is Artin’s famous induction theorem (if \(\chi\) is a rationally represented character of a finite group G then \(| G|_{\chi}\) may be written as a \({\mathbb{Z}}\)-linear combination of permutation characters of G induced from cyclic subgroups). The most general result in this direction is the induction theorem due to S. D. Berman and E. Witt. Chapter 4 uses techniques of algebraic geometry. Chapter 5 is application to exceptional Weyl groups of the methods of the previous chapter.

This book is an interesting application of character theory to finite groups. It gives much information on the mysterious class of Q-groups.

Reviewer: Ya.G.Berkovich

##### MSC:

20C15 | Ordinary representations and characters |

20-02 | Research exposition (monographs, survey articles) pertaining to group theory |

20D20 | Sylow subgroups, Sylow properties, \(\pi\)-groups, \(\pi\)-structure |