Cutting subgroup

From Groupprops
Jump to: navigation, search
This article defines a subgroup property: a property that can be evaluated to true/false given a group and a subgroup thereof, invariant under subgroup equivalence. View a complete list of subgroup properties[SHOW MORE]


This article defines a term that has been used or referenced in a journal article or standard publication, but may not be generally accepted by the mathematical community as a standard term.[SHOW MORE]

History

The historical roots of this term, viz how the term and the concept were developed, are missing from this article. If you have any idea or knowledge, please contribute right now by editing this section. To learn more about what goes into the History section, click here

Definition

Definition with symbols

A subgroup H of G is said to be a cutting subgroup if it is a self-normalizing subgroup and satisfies the further conditions (equivalent subject to the subgroup being self-normalizing):

  • There exists a real representation V of G such that the dimension of V^H is one more than the dimension of V^G (V^H and V^G are the subspaces comprising points fixed pointwise under the action of H and G respectively)
  • There exists a real representation V of G such that V^G has dimension 0 and V^H has dimension 1
  • There exists an irreducible nontrivial real representation V of G such that V^H has dimension 1

References

  • Equivariant deformations of matrices and real representations by Davide L. Ferrario, Pacific Journal of Mathematics, Vol. 196, No. 2, 2000

External links