中文摘要：

当今美国是世界头号数学强国,但在19世纪它的数学还远远落后于欧洲国家。通过留学国外以及自身教育科研机制的改革,美国代数学不断成长壮大。群论作为代数学的一个重要分支,历经科尔、米勒和迪克森等人的不懈努力,在低阶代换群、抽象群、一般线性群以及有限单群方面取得领先世界的研究成果,成为美国代数学领域众多崛起分支的一个典范。

英文摘要：

The mathematics study of America is of the first level in the world today, but America lagged far behind the Europe in the 19 th century. By studying abroad and reforming the education and scientific research system, the Americans have developed algebra into a mighty subject gradually. Through the efforts made by F. N. Cole, G. A. Miller, L. E. Dickson and many other mathematicians, a great achievement ahead of the world has been achieved in the substitution groups and abstract groups of low order, the general linear groups and the finite simple groups. Consequently, group theory, as an important branch of algebra, becomes a striking example of the rising others in the field of American algebra.