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

[naele]

Homework Statement

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

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.