- #1
gentsagree
- 96
- 1
I surely am missing something about the notion of connectedness, and I clarify this by means of an example:
O(n), the orthogonal group, has two subsets with detO=1 and detO=-1. Now, the maximally connected component of O(n) is SO(n), which is the subgroup with detO=1 including the Identity, while the other part is simply a coset (and not a subgroup, as it doesn't, of course, contain I).
Thus, O(n) is NOT a connected group.
I do not understand why, on the other hand, U(n) is said to be connected when it has got, in exactly the same way as O(n), two subsets with detU=1 and detU=-1, where we call the former SU(n).
O(n), the orthogonal group, has two subsets with detO=1 and detO=-1. Now, the maximally connected component of O(n) is SO(n), which is the subgroup with detO=1 including the Identity, while the other part is simply a coset (and not a subgroup, as it doesn't, of course, contain I).
Thus, O(n) is NOT a connected group.
I do not understand why, on the other hand, U(n) is said to be connected when it has got, in exactly the same way as O(n), two subsets with detU=1 and detU=-1, where we call the former SU(n).