This book is about finite groups, but of course infinite groups will turn up from time to time, either surreptitiously, as m the guise of additive or multiplicative subgroups of infinite fields, or quite openly, as in the case of the Lausch group of a Fitting class. The unspoken rule is that all groups are finite, except where they are obviously not! As our title suggests, our groups are also soluble. But here we are less prescriptive and wander quite frequently into insoluble territory to see where the soluble theory leads. Nevertheless, for certain stretches of the narrative, we confine ourselves exclusively to the universe of finite soluble groups. Much of our work is concerned with classes of groups. When these are viewed as sets of isomorphism classes, they become subsets of countable sets, and so, with this interpretation, we are able to talk about sets of classes of groups without challenging the conventions of set theory.

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