Published online by Cambridge University Press: 31 October 2024
The most combinatorially interesting maximal subgroups of M24 are the stabilizers of an octad, a duum, a sextet and a trio. In this chapter we investigate the way in which the stabilizer of one of these objects acts on the others. This involves some basic but fascinating character theory; the approach given here is intended to be self-contained. For each of the four types of object we draw a graph in which each member is joined to members of the shortest orbit of its stabilizer. Thus in the octad graph we join two octads if they are disjoint; we join two dua if they cut one another 8.4/4.8; we join two sextets if the tetrads of one cut the tetrads of the other (22.04)6; and we join two trios if they have an octad in common. A diagram of each of these four graphs is included as is the way in which these graphs decompose under the action of one of the other stabilizers. Each of these graphs is, of course, preserved by M24.
To save this book to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.
Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.
Find out more about the Kindle Personal Document Service.
To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Dropbox.
To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.