Historical definitions of normal subgroup

The first definition of the concept of normal subgroups was given by Galois, with the name décomposition propre.

Translated into English, it states:

In both cases the group of the equation is partitioned by the adjunction into groups such that one passes from one to the other by means of the same substitution; but it is only in the second case that it is certain that these groups have the same substitutions. This is called a proper decomposition.
In other words, when a group

$G$

contains another group

$H$

, then the group

$G$

cam be divided into groups that are obtained by performing the same substitution on the permutations of

$H$

, so that:

$G=H+HS+H{S}^{\prime }+\dots$
It can also be divided into groups with the same substitutions so that
$G=H+TH+T^{\prime }H+\dots$

These two kinds of decompositions do not ordinarily coincide. When they do, the decomposition is said to be proper.

References

Paper references

• Paper:Galois97^{More info}

Textbook references

• The Genesis of the Abstract Group Concept by Hans Wussing, ISBN 0486458687^{More info}, Page 115