# Lie groups and orthogonal group

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 ?

• An analogy: would you consider the nonzero real numbers $\mathbf R^\times$ not to be a Lie group because the sign function is $-1$ on negative numbers? Lie groups do not have to be connected. – KCd Mar 28 at 14:21

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 $$O(n,\mathbb R)$$.
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 $$0$$.