Non-abelian representations of geometries. (English) Zbl 1004.51015
Bannai, Eiichi (ed.) et al., Groups and combinatorics - in memory of Michio Suzuki. Tokyo: Mathematical Society of Japan. Adv. Stud. Pure Math. 32, 301-314 (2001).
In fact, this paper is a very interesting survey paper on geometries for sporadic simple groups. At the same time, as applications, some known results on automorphism groups, covers and universal representations are generalized and improved.
The leading thread is the not necessarily abelian representation of a geometry inside (a subgroup of) its automorphism group. To define this, one needs a geometry every line of which is incident with exactly \(p+1\) points, with \(p\) a prime. Then one embeds the points as distinct subgroups of order \(p\) in a group \(G\) (such that all these groups generate \(G)\) in such a way that the \(p+1\) points of a line correspond to \(p+1\) different subgroups of a subgroup of order \(p^2\). When \(G\) is elementary abelian one gets an ordinary projective embedding. It seems that some series of geometries for sporadic groups admit representations, and for the highest rank member of each family the group \(G\) is non-abelian. The author explains in the paper why this is the case.
It is also shown how one constructs geometries using the idea of a (non-abelian) representation. Many examples are given and much insight in the matter – such as the explanation of the phenomenon mentioned in the previous paragraph – is shared. The last section of the paper is devoted to exploring techniques of calculating the universal representation, in particular the special case \(p=2\) is treated.
For the entire collection see [Zbl 0983.00069].
The leading thread is the not necessarily abelian representation of a geometry inside (a subgroup of) its automorphism group. To define this, one needs a geometry every line of which is incident with exactly \(p+1\) points, with \(p\) a prime. Then one embeds the points as distinct subgroups of order \(p\) in a group \(G\) (such that all these groups generate \(G)\) in such a way that the \(p+1\) points of a line correspond to \(p+1\) different subgroups of a subgroup of order \(p^2\). When \(G\) is elementary abelian one gets an ordinary projective embedding. It seems that some series of geometries for sporadic groups admit representations, and for the highest rank member of each family the group \(G\) is non-abelian. The author explains in the paper why this is the case.
It is also shown how one constructs geometries using the idea of a (non-abelian) representation. Many examples are given and much insight in the matter – such as the explanation of the phenomenon mentioned in the previous paragraph – is shared. The last section of the paper is devoted to exploring techniques of calculating the universal representation, in particular the special case \(p=2\) is treated.
For the entire collection see [Zbl 0983.00069].
Reviewer: H.Van Maldeghem (Gent)
