Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Elementary property of maximal compact subgroup

  1. Feb 4, 2010 #1

    quasar987

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    It is said on wiki* that

    "Maximal compact subgroups are not unique unless the group G is a semidirect product of a compact group and a contractible group, but they are unique up to conjugation, meaning that given two maximal compact subgroups K and L, there is an element g in G such that [itex]gKg^{-1}=L[/itex] – hence a maximal compact subgroup is essentially unique, and people often speak of "the" maximal compact subgroup."

    Why is that so? If the action of G on itself by conjugation were transitive it would be obvious but it isn't, is it?


    *http://en.wikipedia.org/wiki/Maximal_compact_subgroup#Existence_and_uniqueness
     
  2. jcsd
  3. Feb 5, 2010 #2

    quasar987

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    Just to clarify, what I am asking about is the "unique up to conjugation" part, not the first part about uniqueness in the case G is a semidirect product of a compact group and a contractible group.
     
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook