Abstract
For any group, a subgroup is a subset of elements of the original group that is closed under the group operation. That is, the product of any two elements of the subgroup is again an element of the subgroup. For Lie groups, we have the concept of a Lie subgroup. In addition to the closure requirement, the subgroup must also be a submanifold of the group manifold of the original group. It is quite possible to have subgroups of Lie groups that are not Lie subgroups. However, when we talk about the subgroups of a Lie group we will always mean a Lie subgroup. So, for consistency, the group manifold of a discrete group will be thought of as a zero-dimensional manifold. For example, the trivial group has just a single element, the identity element. We will write this group as 0 = {e}; notice that 0 is a subgroup of every group.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 1996 Springer Science+Business Media New York
About this chapter
Cite this chapter
Selig, J.M. (1996). Subgroups. In: Geometrical Methods in Robotics. Monographs in Computer Science. Springer, New York, NY. https://doi.org/10.1007/978-1-4757-2484-4_3
Download citation
DOI: https://doi.org/10.1007/978-1-4757-2484-4_3
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4757-2486-8
Online ISBN: 978-1-4757-2484-4
eBook Packages: Springer Book Archive