In order to claim the rights to the company name. You need to have 10 members in your company. When that happens, the company will disappear from the Create Company page. The staff of the other companies with the same name and letter as yours, will also be presented with the page below where they have to decide if they want to change their company letter, that is, if there's any available, or if they want to just disband their company.
If the company that was claimed should fall below 5 members, due to desertions or discharges. The company will become unclaimed and the company will again appear on the Create Company page for other soldiers to form.