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A well-known theorem of Philip Hall states that if a group  has a nilpotent normal subgroup  such thatG  /   N  ′G/N^{\prime}    is nilpotent, then  itself is nilpotent. We say that a group class  is a  Hall class  if it contains every group  admitting a nilpotent normal subgroup  such thatG  /   N  ′G/N^{\prime}    belongs to . Examples have been given in [F. de Giovanni, M. Trombetti and B. A. F. Wehfritz, Hall classes of groups, to appear] to show that finite-by- groups do not form a Hall class for many natural choices of the Hall class . Although these examples are often linear, our aim here is to prove that the situation is much better within certain natural subclasses of the universe of linear groups.

Hall classes in linear groups