Is there an isomorphism between O(2n) and SO(2n)xZ2?

[naele]

Homework Statement

Is there an isomorphism between
$$O(2n)\simeq SO(2n)\times \mathbb{Z}_2$$
$$O(2n+1)\simeq SO(2n+1)\times \mathbb{Z}_2$$

Homework Equations

First isomorphism theorem

The Attempt at a Solution

I think, if I can show a homomorphism between $SO(2n)\times\mathbb{Z}_2$ and O(2n) then showing whether the kernel is just the identity or not shouldn't be too difficult. But, this is my first time showing a homomorphism with a product of two groups, so I'm unsure how to proceed, namely how do I show that $\phi(S_1S_2)=\phi(S_1)\phi(S_2)$

 

[naele]

Kind of an update, but for the odd case if I use regular matrix multiplication as a homomorphism it seems fairly straightforward, but the even case is still giving me trouble.

 

[fzero]

The odd case is straightforward because an odd-dimensional matrix of unit determinant becomes a matrix of determinant -1 by multiplying by $$-I$$. So any element of $$O(2n+1)$$ can be obtained by multiplying an element of $$SO(2n+1)$$ by either $$\pm I$$.

In the odd case, this doesn't work, but there is a semidirect product structure

$$O(2n)\simeq SO(2n)\ltimes \mathbb{Z}_2 .$$

In this case, we can obtain a matrix with determinant -1 by multiplying an $$SO(2n)$$ matrix by an involution $$R$$ that negates an odd number of columns. Equivalently this is composed of an odd number of reflections in $$\mathbb{R}^{2n}$$. Since $$R$$ is a nontrivial automorphism of $$SO(2n)$$, this is not a direct product.

 

[naele]

I see thanks for the help. Semidirect products are beyond the scope of my class, though at least so far.