Skip to main content Accessibility help

We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.

Close cookie message
×
Hostname: page-component-6bf8c574d5-9nwgx Total loading time: 0 Render date: 2025-02-23T08:44:06.183Z Has data issue: false hasContentIssue false

Appendix: Subgroups of Symmetric Groups

Published online by Cambridge University Press:  06 July 2010

John Swallow
John Swallow
Affiliation:
Davidson College, North Carolina
Get access

Summary

The Subgroups of S4

There are eleven conjugacy classes of the 30 subgroups of S4, as follows:

  1. (1) {〈0〉}, the class consisting of the single subgroup of order 1.

  2. (2) The class of all subgroups of order 2 that are generated by a single transposition, that is, subgroups of the form 〈(ij)〉 for distinct i and j. This class has six elements.

  3. (3) The class of all subgroups of order 2 that are generated by the product of two disjoint transpositions, that is, subgroups of the form 〈(ij)(kl)〉 for distinct i, j, k, and l. This class has three elements.

  4. (4) The class of all subgroups of order 3. These are generated by a 3-cycle and take the form 〈(ijk)〉 for distinct i, j, and k. This class has four elements.

  5. (5) The class of all cyclic subgroups of order 4. These are generated by a 4-cycle and take the form 〈(ijkl)〉 for distinct i, j, k, and l. This class has three elements.

  6. (6) The class of all subgroups of order 4 isomorphic to the Klein 4-group ℤ/2ℤ ⊕ ℤ/2ℤ that are generated by two transpositions. These subgroups have the form 〈(ij), (kl)〉 for distinct i, j, k, and l. This class has three elements.

  7. (7) {〈(12)(34), (13)(24)〉}, the class consisting of the unique subgroup of order 4 isomorphic to the Klein 4-group ℤ/2ℤ ⊕ ℤ/2ℤ that is generated by two products of two transpositions.

  8. (8) The class of all subgroups of order 6. These have the form 〈(ijk), (ij)〉 for distinct i, j, and k. This class has four elements.

  9. […]

Type
Chapter
Information
Exploratory Galois Theory , pp. 197 - 200
Publisher: Cambridge University Press
Print publication year: 2004

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Save book to Kindle

To save this book to your Kindle, first ensure [email protected] 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.

Available formats
×

Save book 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 Dropbox.

Available formats
×

Save book to Google Drive

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.

Available formats
×