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A division of a business or business division (rarely and confusingly called a business sector) is one of the large parts into which a business (company) is divided. The divisions are distinct parts of that business. If these divisions are all part of the same company, then that company is legally responsible for all of the obligations and debts of the division. However in a large organisation, various parts of the business may be run by different subsidiaries, and a business division may include one or many subsidiaries. Each subsidiary is a separate legal entity owned by the primary business or by another subsidiary in the hierarchy. Often a division operates under a separate name and is the equivalent of a corporation or limited liability company obtaining a fictitious name or "doing business as" certificate and operating a business under that fictitious name.
Generally, only an "entity", e.g. a corporation, limited liability company, etc. would have a "division"; an individual operating in this manner would simply be "operating under a fictitious name".
An example of this would be to look at Hewlett Packard (HP), the computer and printer company. HP has several divisions, with the printer division, that makes laser and inkjet printers, being the largest and most profitable division. The divisions of HP, like the Printing & Multifunction division, the Handheld Devices (includes the old calculator) division, the Servers division (mini and mainframe computers), etc., all use the HP brand name. But, Compaq (a part of HP since 2002), operates as a subsidiary, using the Compaq brand name.
Mack Trucks continues to run itself separately, but is a wholly owned subsidiary of AB Volvo. Volvo Trucks also sells trucks under the Volvo name. Within Mack, there is a division named Mack de Venezuela C.A. that does final assembly and sells trucks in South America.
Another more obvious example is that Google Video is a division of Google, and is part of the same corporate entity. But the YouTube video service is a subsidiary of Google because it remains operated as YouTube, LLC, a separate business entity even though it is owned by Google.