Wehlau's conjecture for the dihedral groups

Let [Formula: see text] be a field of characteristic [Formula: see text], [Formula: see text] a finite group whose order is divisible by [Formula: see text] and [Formula: see text] a finite dimensional [Formula: see text]-module. The maximum degree of an indecomposable element of the ring of invariants [Formula: see text] called the Noether number of the representation and is denoted by [Formula: see text]. We consider a conjecture of Wehlau: if [Formula: see text] is the [Formula: see text]-submodule of the [Formula: see text]-module [Formula: see text], then [Formula: see text] We prove that for the dihedral groups of order [Formula: see text] with [Formula: see text] and for the dihedral groups of order [Formula: see text] with [Formula: see text] and an odd positive integer [Formula: see text], the conjecture holds.