On Lee's Introduction to smooth manifold, Page 173, there is a problem wants to show that for $n>1$ $O(n)\cong SO(n) \rtimes O(1)$(which has been shown) is not isomorphic to $SO(n)\times O(1)$ as a Lie group.
He hints that isomorphic groups have isomorphic center. I can only show that for even $n>1$ they don't have isomorphic center, but for odd $n$ I can't do anything about it.
That problem is stated incorrectly. See the correction in my online correction list.