The finite irreducible linear \(2\)-groups of degree \(4\).

*(English)*Zbl 0927.20026
Mem. Am. Math. Soc. 613, 77 p. (1997).

From the abstract and introduction: A linear group of degree \(n\) is a subgroup of \(\text{GL}(n,\mathbb{C})\). Each class of linear groups considered is a union of conjugacy classes. To list the linear groups in a class means to provide a complete list of representatives of their \(\text{GL}(n,\mathbb{C})\) -conjugacy classes.

For a prime \(p\) the problem of listing the finite irreducible linear \(p\)-groups of degree \(n\) (necessarily a power of \(p\)) was considered for \(n=p\) by S. B. Conlon [in J. Aust. Math. Soc., Ser. A 22, 221-233 (1976; Zbl 0338.20024)]. The listing problem for \(n=p^2\) and a general \(n\) is extremely difficult. The purpose of this monograph is to present a solution of the listing problem in the special case \(n=4\). Even in this case substantial difficulties exist.

Each group listed is defined by a generating set of monomial matrices. There are essentially three possibilities for the projection of an irreducible monomial \(2\)-group into the group of all the permutation matrices. The classification problem splits into three cases. Each case may be handled by a general method consisting of three major steps. Techniques for applying the method are developed in detail, so that the other two cases may be dealt with routinely.

These techniques include character theory, a method for drawing the Hasse diagram of the submodule lattice of a direct sum and cohomology theory (with the calculation of \(2\)-cohomology by means of the Lyndon-Hochschild-Serre spectral sequence). Related questions concerning isomorphism between the listed groups, and Schur indices over \(\mathbb{Q}\) are also considered.

One motivation for studying this type of listing problems is the desire in computational group theory for better soluble quotient algorithms. The list given in this monograph serves as a preparatory step toward extending presently available databases.

For a prime \(p\) the problem of listing the finite irreducible linear \(p\)-groups of degree \(n\) (necessarily a power of \(p\)) was considered for \(n=p\) by S. B. Conlon [in J. Aust. Math. Soc., Ser. A 22, 221-233 (1976; Zbl 0338.20024)]. The listing problem for \(n=p^2\) and a general \(n\) is extremely difficult. The purpose of this monograph is to present a solution of the listing problem in the special case \(n=4\). Even in this case substantial difficulties exist.

Each group listed is defined by a generating set of monomial matrices. There are essentially three possibilities for the projection of an irreducible monomial \(2\)-group into the group of all the permutation matrices. The classification problem splits into three cases. Each case may be handled by a general method consisting of three major steps. Techniques for applying the method are developed in detail, so that the other two cases may be dealt with routinely.

These techniques include character theory, a method for drawing the Hasse diagram of the submodule lattice of a direct sum and cohomology theory (with the calculation of \(2\)-cohomology by means of the Lyndon-Hochschild-Serre spectral sequence). Related questions concerning isomorphism between the listed groups, and Schur indices over \(\mathbb{Q}\) are also considered.

One motivation for studying this type of listing problems is the desire in computational group theory for better soluble quotient algorithms. The list given in this monograph serves as a preparatory step toward extending presently available databases.

Reviewer: G.Călugăreanu (Cluj-Napoca)

##### MSC:

20G20 | Linear algebraic groups over the reals, the complexes, the quaternions |

20D15 | Finite nilpotent groups, \(p\)-groups |

20E07 | Subgroup theorems; subgroup growth |

20C15 | Ordinary representations and characters |

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

20E45 | Conjugacy classes for groups |