Semidirect Product

Let $\Gamma$ and $H$ be groups and let $\Gamma$ act on $H$ by automorphisms ($\gamma:h\mapsto {}^\gamma h$).

The semidirect product $\Gamma\ltimes H$ is the set $\Gamma\times H$ with multiplication $(\gamma,h)(\gamma',h')=(\gamma\gamma',\,h\cdot {}^\gamma h').$

In the letter: the $L$-group is a semidirect product of a Galois group with the dual group .