Historical definitions of normal subgroup

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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:
$G = H + HS + HS' + \dots$
It can also be divided into groups with the same substitutions so that
$G = H + TH + T'H + \dots$
These two kinds of decompositions do not ordinarily coincide. When they do, the decomposition is said to be proper.