Lie groups and orthogonal group

Question

Are orthogonal groups are lie groups? I think parameter space points corresponds to elements with determinant -1 break analytic property of lie groups , what is the general condition to check a group is lie group or not ?

Answer 1

Yes, orthogonal groups are Lie groups. Since, $O(n,\mathbb R)$ consists of two copoes of $SO(n,\mathbb R)$, if $SO(n,\mathbb R)$ is a Lie group, then so is $O(n,\mathbb R)$.

Answer 2

Orthogonal groups can be defined over arbitrary fields $K$ as the subgroup of the general linear group $GL_n(K)$ given by $$ \operatorname {O} (n,K)=\left\{Q\in \operatorname {GL} (n,K)\;\left|\;Q^{\mathsf {T}}Q=QQ^{\mathsf {T}}=I\right.\right\}. $$ Now for example, for a finite field we obtain a finite group, which is also a compact discrete Lie group of dimension $0$.

Real orthogonal groups are real linear groups and hence real Lie groups.