altmann math

Relation between Altmann’s definition of semi-direct product and the one in Wikipedia

First off, notice that Altmann’s says that \(\mathbf{S}\) (the set of coset representatives) is a group. Secondly, \(\mathbf{H}\triangleleft\mathbf{S}\) is equivalent (as can be easiliy shown) to being a normal subgroup of \(\mathbf{G}\).
Thus, \(\mathbf{H}\) is a normal subgroup of \(\mathbf{G}\). Therefore, we recover the original meaning of \(\mathbf{G} = \mathbf{H} \rtimes \mathbf{S}\).