Elementary property of maximal compact subgroup

  • Thread starter quasar987
  • Start date
  • #1
quasar987
Science Advisor
Homework Helper
Gold Member
4,780
12
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
 

Answers and Replies

  • #2
quasar987
Science Advisor
Homework Helper
Gold Member
4,780
12
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.
 

Related Threads on Elementary property of maximal compact subgroup

Replies
2
Views
2K
  • Last Post
Replies
3
Views
7K
Replies
3
Views
2K
Replies
1
Views
2K
Replies
6
Views
2K
Replies
1
Views
2K
  • Last Post
Replies
1
Views
1K
  • Last Post
Replies
1
Views
605
Replies
2
Views
2K
  • Last Post
Replies
8
Views
2K
Top