Semi-direct products

The semi direct product and the weak direct product have been defined in Section 1.5. 

Theorem. Let G be a finite group scheme over a perfect field. Then G is the semi-direct product of G° and n0 G. 

Throughout the present paper, the concepts of direct and semi-direct products will be used. Let us define them next. For this purpose, let us consider two groups, I and G, which have no element in common except for the identity. 

As a result, the isodynamic subgroup, /, may be expressed as the semi-direct product ... 

Notice that the full NRG s of this series of molecular systems may not be written as a semi-direct product of a restricted NRG for the internal motions and a symmetry point subgroup for the external rotation, as Altmann presented [10], because of the coupling between both motions. The full NRG s, however, possess the same group structure as the Longuet-Higgins Molecular Symmetry Group, in all the cases. 

N x H into a group scheme which is the semi-direct product of N and H. 

Scheme 44 Schur s lemma 63 Semi-direct product 19 Semi-invariant element 34 Semisimple group 97 Separable algebra 47 Separable matrix 54 Sheaf 43...

Brailov, A. V. A series of completely integrable Hamiltonian systems on a semi-direct product 5 0(n) x iE". In Geometry and Topology in Global Nonlinear Problems. Voronezh, Voronezh Univ. Press, (1984), 145-148. 

The existence of at least one invariant mean characterizes a class of groups, the amenable groups, which include translation groups R", rotation groups O", and Euclidian groups E . Rem. the reason is R" abelian, G compact, and E semi-direct product of R and O") However, the Loren tz group is not amenable. 

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