The Monster group and Majorana involutions.

*(English)*Zbl 1205.20014
Cambridge Tracts in Mathematics 176. Cambridge: Cambridge University Press (ISBN 978-0-521-88994-0/hbk). xiii, 252 p. (2009).

One of the most interesting sporadic simple groups, the Monster group \(M\), was discovered by B. Fischer and R. Griess then gave the first existence proof for this group. The book under review is just about this group. The group \(M\) has a class of involutions, so called \(2B\)-involutions, such that the centralizer \(G_1\) is an extension of an extraspecial \(Q\) of order \(2^{55}\) by \(Co_1\), where \(Co_1\) acts on \(Q/Z(Q)\) as on the Leech lattice modulo 2. Furthermore there is a fours group of such involutions such that the normalizer \(G_2\) of this fours group is an extension of a group of order \(2^{35}\) by \(\Sigma_3\times M_{24}\) and there is an elementary Abelian group of order eight consisting entirely of involutions of type \(2B\) whose normalizer \(G_3\) is an extension of a group of order \(2^{39}\) by \(3\Sigma_6\times L_3(2)\). In fact with these groups one can build up the famous tilde geometry for \(M\). In [Geometry of sporadic groups. I: Petersen and tilde geometries. Encyclopedia of Mathematics and Its Applications 76. Cambridge: Cambridge University Press (1999; Zbl 0933.51006) and Geometry of sporadic groups II: Representations and amalgams. Encyclopedia of Mathematics and Its Applications. 91. Cambridge: Cambridge University Press (2002; Zbl 0992.51006)] A. A. Ivanov and S. V. Shpectorov have shown that \(M\) is a universal completion of this amalgam \((G_1,G_2,G_3)\). This is the point of view \(M\) is considered in this book. Hence a Monster group is just a faithful completion of the Monster amalgam \((G_1,G_2,G_3)=\mathcal M\). The author shows that a nontrivial completion exists and is unique and in fact the universal completion is the Monster group. This implies existence and uniqueness of the Monster. This follows the line as in M. Aschbacher’s book [Sporadic groups. Cambridge Tracts in Mathematics. 104. Cambridge: Cambridge University Press (1994; Zbl 0804.20011)], which is based on treatments of Griess, Conway, Tits and Norton. This all will be done within 7 chapters.

Chapter 1, \(M_{24}\) and all that: Here we see \(M_{24}\), the Golay-code, the Parker loop and all the basic ingredients which will be used later on.

Chapter 2, The Monster amalgam \(\mathcal M\): Here we see the structure of the groups \(G_1\), \(G_2\), \(G_3\) and the existence and uniqueness of \(\mathcal M\) is shown.

Chapter 3, 196883 – representation of \(\mathcal M\): Here a minimal representation \(\varphi\) of \((G_1,G_2)\) is constructed and extended to \(\mathcal M\) showing that there is a faithful completion.

Chapter 4, 2-local geometries: Here the famous tilde geometry for \(M\) is constructed.

Chapter 5, Griess algebra: Here the Griess algebra \(V\) is constructed using Norton’s approach.

Chapter 6, Automorphisms of Griess algebra: Here the multiplication in \(C_V(Z(G_1))\) is given and shown that \(C_{\operatorname{Aut}(V)}(Z(G_1))=G_1\). From this it follows that \(\varphi(G)\) is a finite simple group.

In the final Chapter 9 \(G\cong\varphi(G)\) is shown, based on the simple connectedness of the tilde geometry. This implies that \(M\) in the unique faithful completion of the Monster amalgam.

In between in Chapter 7 some important subgroups of \(M\) like \(3M(24)\), \(2BM\), \(Th\) and \(HN\) are constructed.

Chapter 8 is devoted to what the author calls Majorana involutions. In a vertex operator algebra \(V=\bigoplus^\infty_{n=0}V_n\), \(V_2\) is a commutative not associative algebra, called Griess algebra. The most famous example is the Griess algebra related to the Monster group. In this algebra idempotent elements, so called Ising elements, play an important role. The Majorana elements are just a little weaker than Ising elements. One Ising element gives birth to an involutive automorphism in any general setting. S. Sakuma [Int. Math. Res. Not. 2007, No. 9, Article ID rnm030 (2007; Zbl 1138.17013)] finally classified all subalgebras generated by two Ising elements. They are in some bijection to the nine orbits of pairs of involutions of type \(2A\) (with centralizer \(2BM\)) in \(M\). Hence it is expected that they might become relevant for investigating the Monster in future. So in Chapter 8 the Majorana calculus is introduced and algebras which are generated by two Majorana elements are investigated. Recently the author, D. V. Pasechnik, Á. Seress and S. Shpectorov [J. Algebra 324, No. 9, 2432-2463 (2010; Zbl 1257.20011)] have classified all subalgebras which are generated by three Majorana elements with the relation of type \(A_3\), corresponding to certain \(\Sigma_4\)-subgroups of \(M\).

This book contains the basic knowledge on the Monster group in a very accessible way. Some results are published in this book for the first time. Many are not even easily found in literature. Hence the book is a very good source for any group theorist who is interested in sporadic simple groups.

Chapter 1, \(M_{24}\) and all that: Here we see \(M_{24}\), the Golay-code, the Parker loop and all the basic ingredients which will be used later on.

Chapter 2, The Monster amalgam \(\mathcal M\): Here we see the structure of the groups \(G_1\), \(G_2\), \(G_3\) and the existence and uniqueness of \(\mathcal M\) is shown.

Chapter 3, 196883 – representation of \(\mathcal M\): Here a minimal representation \(\varphi\) of \((G_1,G_2)\) is constructed and extended to \(\mathcal M\) showing that there is a faithful completion.

Chapter 4, 2-local geometries: Here the famous tilde geometry for \(M\) is constructed.

Chapter 5, Griess algebra: Here the Griess algebra \(V\) is constructed using Norton’s approach.

Chapter 6, Automorphisms of Griess algebra: Here the multiplication in \(C_V(Z(G_1))\) is given and shown that \(C_{\operatorname{Aut}(V)}(Z(G_1))=G_1\). From this it follows that \(\varphi(G)\) is a finite simple group.

In the final Chapter 9 \(G\cong\varphi(G)\) is shown, based on the simple connectedness of the tilde geometry. This implies that \(M\) in the unique faithful completion of the Monster amalgam.

In between in Chapter 7 some important subgroups of \(M\) like \(3M(24)\), \(2BM\), \(Th\) and \(HN\) are constructed.

Chapter 8 is devoted to what the author calls Majorana involutions. In a vertex operator algebra \(V=\bigoplus^\infty_{n=0}V_n\), \(V_2\) is a commutative not associative algebra, called Griess algebra. The most famous example is the Griess algebra related to the Monster group. In this algebra idempotent elements, so called Ising elements, play an important role. The Majorana elements are just a little weaker than Ising elements. One Ising element gives birth to an involutive automorphism in any general setting. S. Sakuma [Int. Math. Res. Not. 2007, No. 9, Article ID rnm030 (2007; Zbl 1138.17013)] finally classified all subalgebras generated by two Ising elements. They are in some bijection to the nine orbits of pairs of involutions of type \(2A\) (with centralizer \(2BM\)) in \(M\). Hence it is expected that they might become relevant for investigating the Monster in future. So in Chapter 8 the Majorana calculus is introduced and algebras which are generated by two Majorana elements are investigated. Recently the author, D. V. Pasechnik, Á. Seress and S. Shpectorov [J. Algebra 324, No. 9, 2432-2463 (2010; Zbl 1257.20011)] have classified all subalgebras which are generated by three Majorana elements with the relation of type \(A_3\), corresponding to certain \(\Sigma_4\)-subgroups of \(M\).

This book contains the basic knowledge on the Monster group in a very accessible way. Some results are published in this book for the first time. Many are not even easily found in literature. Hence the book is a very good source for any group theorist who is interested in sporadic simple groups.

Reviewer: Gernot Stroth (Halle)

##### MSC:

20D08 | Simple groups: sporadic groups |

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

20C34 | Representations of sporadic groups |

17B69 | Vertex operators; vertex operator algebras and related structures |

51E24 | Buildings and the geometry of diagrams |