Next to the concept of a function, which is a most important concept pervading the whole of mathematics, the concept of a group is of the greatest significance in the various branches of mathematics and in its applications. The group concept is not any more difficult to appreciate than the function concept; indeed one can more easily become familiar with this concept during the early stages of a mathematical education than with the subject-matter of elementary mathematics.Every pupil in a senior class of a grammar school who enjoys doing mathematics is capable of grasping the idea of a group if he is interested and industrious. And so this book has been written in the first place for the mathematically inclined pupils in the senior classes in the grammar school, but also for those who teach mathematics to the senior or to the advanced level. As regards the character of the exposition, I have been at pains to introduce no concepts without illustrating them by means of simple examples, for the most part geometrical.The group concept has played a central part in the development of mathematical thought. At the same time, many aspects of the theory of groups are simple enough for the interested grammar school student to appreciate and absorb. We believe that this book by the eminent mathematician P. S. Alexandroff is especially suited for introducing the subject to the young student. The material presented is of fundamental importance and is developed in a clear and rigorous fashion, and the book is particularly noteworthy on accountof the wealth of illustrative examples which are included.We have added some exercises at the end of each chapter, and some references which we hope will be useful to the English-speaking student. We have also made a few minor alterations and additions to the actual text, some of which were made necessary by English usage,and we have corrected some minor errors.