What deployment method did you use to deploy the SEP clients?
Did you use a common system image to deploy, such as utilizing Norton Ghost?
Were the clients installed as User Mode or Computer Mode?
Are you using Active Directory synchronization?
Is the SEPM using an SQL database or embedded, if SQL is the database hosted on the same server as the SEPM console?
If you view the clients locally on their respective systems, does the SEP client shield icon display a green dot?
If you open the SEP client and view Help and Support>Troubleshooting does it show the SEPM server name or IP Address and the correct group name?
If you run a search for clients from the SEPM clients page for a specific computer bane, does it return found?
Regarding your attempt to reinstall, did you simply attempt to push a new install package to the SEP client from the SEPM? If so, most likely the reason it appeared back in the same group is due to the default communications settings for install packages in the SEPM. By default, they are not configured to overwrite clients' existing communications settings.
You can create custom installation settings to change this behavior. Navigate to Admin>Install Packages in the SEPM console. Select Client Install Settings on the upper left and choose Add Client Install Settigns under Tasks. In the window that opens you will see an option at the bottom for Upgrade settings. Ticking the radio button to Remove all previous logs and policies will configure the install to overwrite the existing settings. This should move the client to a new group.
Let us know if this helps out.
Regards.