The Federal Home Loan Bank Board, in an important move toward deregulation, yesterday granted three savings and loan associations, including Perpetual American Federal here, authority to set up a jointly owned brokerage subsidiary that could buy and sell stocks, bonds and mutual funds.

Dan McConnell, chairman of the new firm, said that pilot "investment centers" will begin operating in some Perpetual offices--as well as at savings and loans in California, New Jersey and Florida--by September.

About 340 savings and loans have expressed interest in subscribing to the services provided by the brokerage firm. If that many associations subscribe and put "investment centers" in just two of their branches (the average savings and loan has seven branches), the new firm would have as large a retail brokerage network as Merrill Lynch Pierce Fenner & Smith, by far the nation's biggest today.

The establishment of brokerage offices in savings and loan associations would create something very close to the one-stop financial shopping that the financial services industry has been seeking for the past few years.

That quest has triggered a wave of mergers uniting brokerage firms and such companies as as Sears, American Express and Prudential Insurance. The giant Bank of America is trying to acquire the discount broker Charles E. Schwab & Co.

Each new merger puts another crack into the once seemingly impenetrable wall between the banking industry and the securities industry.

The brokerage industry strongly opposed the move by the savings and loans, arguing that it was such a major change that Congress, not a regulatory agency, should pave the way. The bank board has been searching for ways to expand the money-making abilities of the severely battered savings and loan industry to contain the wave of failures that have swept the industry.

The Senate Banking Committee has been studying the changes in the financial services industry and the Reagan administration has proposed a bill to define how banks, brokers and other financial services firms might tread on each other's once well-defined turf.

Edward I. O'Brien, president of the Securities Industry Association, said he was disappointed with the Bank Board's unanimous decision, but that the trade association has yet to decided whether to try to block the move in court.

Besides Perpetual, the bank board approved Coast Federal in Sarasota, Fla., and California Federal in Los Angeles as founders of the brokerage firm. City Federal in Elizabeth, N.J., has its application pending before the Federal Home Loan Bank in New York.

McConnell said about 30 savings and loans will be owners of a new firm, the Savings Association Financial Corp., which will be the parent company of the brokerage firm, Savings Association Investment Securities. The S&Ls will own shares in the parent company through their service corporations, wholly owned subsidiaries that today offer products, such as insurance, to customers.