Historical definitions of group

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This article lists historical definitions of the following term: group

Refer history of groups for more about the history of the definition and concept of group, and funky definitions of group for a list of weird definitions of group.

Definitions using permutations


In a paper in 1869 titled Commentaire sur Galois, Jordan defines a group as follows:

A system of substitutions forms a group if the product of any two substitutions of the system is again a member of the system.

In his book Traité des substitutions, Jordan defines a permutation (he calls it a substitution) and then goes on to define a group:

We will say that a system of substitutions form a group if the product of any two substitutions in the system is itself in the system.

Definitions in the abstract setting

Cayley (1854)

Cayley attempted the first general definition of a group, in his paper On the theory of groups, as depending on the symbolic equation \theta^n = 1. This definition is not precisely the same as the modern definition, though it is equivalent. Cayley states:

A set of symbols 1, \alpha, \beta, \dots, all of them different, and such that the product of any two of them (no matter in what order), or the product of any one of them into itself, belongs to the set, is said to be a group. These symbols are not in general convertible (commutative) but associative, it follows that if the entire group is multiplied by any one of the symbols, either as further or nearer factor (left or right), the effect is simply to reproduce the group.

This definition is today summarized as saying that a group is an associative quasigroup. It is equivalent to the modern definition of group. Further information: Associative quasigroup equals group

Weber (1882)

Heinrich Weber gave this definition in a paper on quadratic forms.

A system G of h arbitrary elements \theta_1, \theta_2, \dots, \theta_h is called a group of degree h if it satisfies the following conditions:

I. By some rule which is designated as composition or multiplication, from any two elements of the same system one derives a new element of the same system.
II. It is always true that (\theta_r \theta_s)\theta_t = \theta_r(\theta_s\theta_t).

III. From \theta \theta_r = \theta \theta_s it follows that \theta_r = \theta_s.

Weber's definition is equivalent to the modern definition of finite group.


Paper references

  • On the theory of groups, as depending on the symbolic equation \theta^n = 1 by Arthur Cayley, Philosophical Magazine, (Year 1854): This paper is the first correct attempt at a definition of group, though the definition is expressed differently from the modern definition. See the definition here.More info
  • Commentaire sur Galois by Camille Jordan, Math. Ann. 1, Page 141-160, 1869.More info

Textbook references

  • Traité des substitutions et des équations algébriques by Camille Jordan, published 1870: This was a definitive book on groups and permutations for the time, and continued to be an important reference for the next three decades.Full text on Google BooksMore info