Business: facing demands to shoulder more of society's problems.; America's Management Challenge - Capitalizing on Change, by William B. Miller. Radnor, Pa.: Chilton Book Company. 318 pp. $16.50.

WHAT should be the role of big business in carrying out public policy? Corporate good citizens today are expected to be much more than just law abiding. Special-interest groups and the public in general are asking business to shoulder the burdens of change.

Perhaps the most significant aspect of William B. Miller's book, ''America's Management Challenge,'' is his long description of the relationship between business and society. The text details the evolution of public involvement with corporate America. And the author, in so many words, delivers the startling message that an increasingly pluralistic people expects business to take a larger role in solving the problems of society.

Miller, formerly a partner the Touche Ross & Co. management consulting firm and now a professor at the University of California, Berkeley, counts special-interest groups, regulators and government agencies, the courts, legislators, and employees as being among the people expecting business to find and apply solutions for all types of concerns. These concerns include environmental problems; age, race, and sex discrimination; consumer vulnerability; and poverty.

Miller's point is that the trend of modern history is toward more legal and moral demands on American industry. Capitalists are to be responsible to the communities in which they operate. We all know this, of course, but Miller argues energetically that these changes in the role of industry are, for the most part, salutary and should be anticipated, welcomed, and taken advantage of by tomorrow's captains of industry.

The author also attempts to take on virtually all the major challenges facing business people in the United States at all levels in dozens of industries. Some readers might feel these topics are treated rather superficially. More than 20 distinct industries are discussed in about 60 pages. Miller outlines the major challenge for each and offers a key to managing it successfully in the decades to come.

At the risk of oversimplifying the oversimplified, here are some examples:

* Steel. Problem: low average returns and outmoded production facilities and management techniques. Answer: cost control, specialization, management innovation, and more risk-taking.

* Rubber. Problem: too much competition, forcing low returns on equity, exacerbated by foreign competition that tends to lead the way in product innovation. Answer: specialization and cost control.

* Television manufacturing. Problem: foreign competition from the Far East and inadequate tariff protection. Answer: to avoid getting tied up in court fighting the Japanese, as Zenith has, but concentrate on low-cost production, as RCA has.

* Movie theaters. Problem: new media delivering the product, i.e., television , cable television, cassette, and videodisc players. Answer: to get financially involved in the new technology; promote the advantages of theatergoing; cater to movie buffs, the 25 percent of the population that buys 85 percent of the tickets.

* Aluminum. Problem: market saturation, forcing low growth in the future. Answer: not so much to be found in the long term through new products and markets as through cost control in the short term.

* Supermarkets. Problem: large chains having gone about as far as they can go in obtaining market share, limitingfuture growth possibilities. Answer: super stores (goods and services other than groceries), gourmet or specialty food stores, and cost reduction via automated checkout.

* Transportation. Problem: an ''environment of increased competition'' brought on by deregulation. Answer: low-cost operations resulting from economies of scale after mergers and acquisitions.

* Truck manufacturing. Problem: railroad deregulation, the price of diesel fuel, market saturation, and greater foreign competition. Answer: mergers and up-marketing of the product line.

Miller admits that designing strategy for these and other industries is not as easy as his brief synopses make it sound. He is careful to point out that strategic planning is a highly individualistic need; that is, each individual company faces a different configuration of opportunities.

Almost everything you can think of that needs to be considered in guiding an American business out of the mega problem of America's declining position in the world market is touched upon.

Beyond the many insights, the book does tend to leave one wondering if America's management challenge is not just too big a subject for one book - especially one apparently written for readers ranging from MBA candidates to Fortune 500 chief executives. It is witty and easy to read and Miller actually avoids buzzwords. The wisdom is shared in such a way as to be accessible to almost anyone who believes, as Miller does, that American businesses can and must be successful - that is, profitable in the decades to come - and that this success can be aided rather than hindered by the swirling changes of the modern world.