Right of Transfer of Contract?

What is the name of your state (only U.S. law)? Georgia

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If I have a contract with Company A, can they legally transfer that contract to Company B without my knowledge or consent, even though the contract does not stipulate that they have the right of transfer?

It depends. While most contracts can be assigned, some cannot. What was the contract?

This is a security alarm monitoring contract.

That would not be of a sufficiently personal nature to not be assignable. They can transfer it.