Update SharePoint Group Owner using Powershell for SharePoint 2007

Thanks for the response.  Looks like the code you provided is used for adding new groups.  In my situation, I'm trying to iterate all SharePoint Groups within my site and set the owner to an existing SharePoint Group within the site.  I did try to leverage some of your code and came up with the following.  

[System.Reflection.Assembly]::LoadWithPartialName("Microsoft.SharePoint")
$site = New-Object Microsoft.SharePoint.SPSite("https://mysharepointsite/sites/sitename")
$web = $site.OpenWeb()

$groups = $web.sitegroups
$groupowner = $site.RootWeb.AllUsers["Existing SharePoint Group"];

foreach ($grp in $groups){
"Group: " + $grp.name
"Name:" + $grp.Owner.Name
 $grp.Owner.Name = $groupowner
$grp.Update()
 }

$web.Update()
$site.Dispose()

In doing so, I got the following error:
Exception setting "Name": "The parameter Name cannot be empty or bigger than 255 characters."

This tells me that this variable isn't getting set properly:
$groupowner = $site.RootWeb.AllUsers["Existing SharePoint Group"];

Any thoughts on what I could be doing wrong?

I granted you the points.  While I didn't use any of your code, it got me to think that I couldn't assign a Group Owner a string, but rather a Site Group object.

Thanks!!