Overgroups of Sylow subgroups in sporadic groups.

*(English)*Zbl 0585.20005
Mem. Am. Math. Soc. 343, 235 p. (1986).

The author studies the overgroups of a Sylow p-subgroup T of a sporadic simple group G, where \(p^ 2| | G|\). He associates a geometric structure to the collection of overgroups which might be helpful in the study of the subgroup structure and the permutation representations of G. For this purpose the author defines a so called well behaved p-basis, which is a certain family \({\mathcal F}=(G_ i: i\in I)\) of overgroups of T with nice properties. One of these properties is that the set of all intersections of the \(G_ i\) is just the set of all overgroups of \(\cap_{i\in I}G_ i\). These intersections are called the parabolics of G. In the case of a group of Lie type in characteristic p these are exactly the wellknown parabolics.

One of the central results in this paper is to show that with the exceptions of \(M_{11}\) and \(M_{23}\) any sporadic simple group possesses a well behaved p-basis for a suitable prime p. In fact the paper contains a lot of very detailed results about the subgroup structure of the finite simple groups (not only the sporadic groups). Many of them have been known before, but many of them have never been in print before. Nevertheless the spirit of the whole paper is not to establish the result by treating the groups individually. The author tries to set up a kind of theory to prove theorems about simple groups in a general way. From this it follows that many theorems are stated for simple groups in a certain class and not just for sporadic groups. It is not enough space to state any of them.

To give a flavour what is going on I will sketch the following. The most important class is the class of groups of GF(p)-type, which are groups containing a so called large symplectic p-subgroup Q. This class is important because most of the sporadic groups and most of the finite groups of Lie-type over the prime-field GF(p) are in this class. Then one of the basic results is the determination of \(<Q^ G>\) in an almost simple group of GF(p)-type.

This book is highly recommended to anybody who is interested in this area. But if one tries to use it for finding a special property of a special group one will get lost. Such a long paper with so many different results should have some help for a reader who is interested in some of the aspects of this paper but does not have the time to read anything.

One of the central results in this paper is to show that with the exceptions of \(M_{11}\) and \(M_{23}\) any sporadic simple group possesses a well behaved p-basis for a suitable prime p. In fact the paper contains a lot of very detailed results about the subgroup structure of the finite simple groups (not only the sporadic groups). Many of them have been known before, but many of them have never been in print before. Nevertheless the spirit of the whole paper is not to establish the result by treating the groups individually. The author tries to set up a kind of theory to prove theorems about simple groups in a general way. From this it follows that many theorems are stated for simple groups in a certain class and not just for sporadic groups. It is not enough space to state any of them.

To give a flavour what is going on I will sketch the following. The most important class is the class of groups of GF(p)-type, which are groups containing a so called large symplectic p-subgroup Q. This class is important because most of the sporadic groups and most of the finite groups of Lie-type over the prime-field GF(p) are in this class. Then one of the basic results is the determination of \(<Q^ G>\) in an almost simple group of GF(p)-type.

This book is highly recommended to anybody who is interested in this area. But if one tries to use it for finding a special property of a special group one will get lost. Such a long paper with so many different results should have some help for a reader who is interested in some of the aspects of this paper but does not have the time to read anything.

Reviewer: G.Stroth

##### MSC:

20D05 | Finite simple groups and their classification |

20D08 | Simple groups: sporadic groups |

20D30 | Series and lattices of subgroups |

20B25 | Finite automorphism groups of algebraic, geometric, or combinatorial structures |