Algebra and Applications

Group Identities on Units and Symmetric Units of Group Rings

Authors: Lee, Gregory T

  • A useful reference for the expert
  • A helpful introduction to the subject for graduate students
  • Presents up-to-date research in a comprehensive and unified manner
see more benefits

Buy this book

eBook $109.00
price for USA (gross)
  • ISBN 978-1-84996-504-0
  • Digitally watermarked, DRM-free
  • Included format: PDF
  • ebooks can be used on all reading devices
  • Immediate eBook download after purchase
Hardcover $139.00
price for USA
  • ISBN 978-1-84996-503-3
  • Free shipping for individuals worldwide
  • Usually dispatched within 3 to 5 business days.
Softcover $139.00
price for USA
  • ISBN 978-1-4471-2589-1
  • Free shipping for individuals worldwide
  • Usually dispatched within 3 to 5 business days.
About this book

Let FG be the group ring of a group G over a field F. Write U(FG) for the group of units of FG. It is an important problem to determine the conditions under which U(FG) satisfies a group identity. In the mid 1990s, a conjecture of Hartley was verified, namely, if U(FG) satisfies a group identity, and G is torsion, then FG satisfies a polynomial identity. Necessary and sufficient conditions for U(FG) to satisfy a group identity soon followed.

Since the late 1990s, many papers have been devoted to the study of the symmetric units; that is, those units u satisfying u* = u, where * is the involution on FG defined by sending each element of G to its inverse. The conditions under which these symmetric units satisfy a group identity have now been determined.

This book presents these results for arbitrary group identities, as well as the conditions under which the unit group or the set of symmetric units satisfies several particular group identities of interest.

Reviews

From the reviews:

“This book is concerned with one of the main directions in the study of group rings, namely, questions around group identities satisfied by units … . This is a nicely written book, understanding of which requires familiarity with groups and rings on the level of introductory graduate courses. Its publication is timely, taking into account the progress made in the considered theme, and I believe that it is both interesting and useful for everyone with an eye to group rings.” (Michael Dokuchaev, Mathematical Reviews, Issue 2012 d)

“The author undertakes the laborious task of expounding the … structure of unit groups of group algebras and of units of group algebras symmetric to the classical involution … . The likely reader … may not only be the specialist but the scholar whose research interests have connections with either group or ring theory … . The extensive bibliography of items relating to topics discussed allows the curious reader to gain an even deeper insight to a specific question by consulting the literature.” (János Kurdics, Zentralblatt MATH, Vol. 1203, 2011)


Table of contents (7 chapters)

  • Group Identities on Units of Group Rings

    Lee, Gregory T.

    Pages 1-43

  • Group Identities on Symmetric Units

    Lee, Gregory T.

    Pages 45-75

  • Lie Identities on Symmetric Elements

    Lee, Gregory T.

    Pages 77-101

  • Nilpotence of $$\mathcal{U}(FG)$$ and $${\mathcal{U}}^{+}(FG)$$

    Lee, Gregory T.

    Pages 103-135

  • The Bounded Engel Property

    Lee, Gregory T.

    Pages 137-147

Buy this book

eBook $109.00
price for USA (gross)
  • ISBN 978-1-84996-504-0
  • Digitally watermarked, DRM-free
  • Included format: PDF
  • ebooks can be used on all reading devices
  • Immediate eBook download after purchase
Hardcover $139.00
price for USA
  • ISBN 978-1-84996-503-3
  • Free shipping for individuals worldwide
  • Usually dispatched within 3 to 5 business days.
Softcover $139.00
price for USA
  • ISBN 978-1-4471-2589-1
  • Free shipping for individuals worldwide
  • Usually dispatched within 3 to 5 business days.
Loading...

Recommended for you

Loading...

Bibliographic Information

Bibliographic Information
Book Title
Group Identities on Units and Symmetric Units of Group Rings
Authors
Series Title
Algebra and Applications
Series Volume
12
Copyright
2010
Publisher
Springer-Verlag London
Copyright Holder
Springer-Verlag London Limited
eBook ISBN
978-1-84996-504-0
DOI
10.1007/978-1-84996-504-0
Hardcover ISBN
978-1-84996-503-3
Softcover ISBN
978-1-4471-2589-1
Series ISSN
1572-5553
Edition Number
1
Number of Pages
XII, 196
Topics