We prove a result showing the connection between induced orthogonal representations and corestrictions of group extensions derived from their Clifford groups. By the means of the corestriction map we then obtain new obstructions to the μ p -embedding problems given by the group extensions of the modular p-group, and to one μ 2 -embedding problem given by a group extension of the dihedral group.

Let G be a finite group and k be a field. Let G act on the rational function field k(x g : g ∈ G) by k-automorphisms defined by g · x h = x gh for any g, h ∈ G. Noether's problem asks whether the fixed field k(G) = k(x g : g ∈ G) G is rational (i.e. purely transcendental) over k. Theorem 1. If G is a group of order 2 n (n ≥ 4) and of exponent 2 e such that (i) e ≥ n − 2 and (ii) ζ 2 e−1 ∈ k, then k(G) is k-rational. Theorem 2. Let G be a group of order 4n where n is any positive integer (it is unnecessary to assume that n is a power of 2). Assume that (i) char k = 2, ζ n ∈ k, and (ii) G contains an element of order n. Then k(G) is rational over k, except for the case n = 2m and G ≃ C m ⋊ C 8 where m is an odd integer and the center of G is of even order (note that C m is normal in C m ⋊ C 8 ) ; for the exceptional case, k(G) is rational over k if and only if at least one of −1, 2, −2 belongs to (k × ) 2 . is rational (=purely transcendental) over k. It is related to the inverse Galois problem, to the existence of generic G-Galois extensions over k, and to the existence of versal Gtorsors over k-rational field extensions [Sw; Sa1; GMS, 33.1, p.86]. Noether's problem for abelian groups was studied extensively by Swan, Voskresenskii, Endo, Miyata and Lenstra, etc. The reader is referred to Swan's paper for a survey of this problem [Sw].On the other hand, just a handful of results about Noether's problem are obtained when the groups are not abelian. It is the case even when the group G is a p-group. The reader is referred to [CK; Ka3; HuK; Ka6] for previous results of Noether's problem for p-groups. In the following we will list only those results pertaining to the 2-groups.Theorem 1.1 (Chu, Hu and Kang [CHK; Ka2]) Let k be any field. Suppose that G is a non-abelian group of order 8 or 16. Then k(G) is rational over k, except when char k = 2 and G = Q 16 , the generalized quaternion group of order 16. When char k = 2 and G = Q 16 , then k(G) is also rational over k provided that k(ζ 8 ) is a cyclic extension over k where ζ 8 is a primitive 8-th root of unity.Theorem 1.2 (Serre [GMS, Theorem 34.7]) If G = Q 16 , then É(G) is not stably rational over É; in particular, it is not rational over É.We don't know the answer whether k(G) is rational over k or not, if G = Q 16 and k is any field other than É such that k(ζ 8 ) is not a cyclic extension of k. The reader is referred to [CHKP; CHKK] for groups of order 32 and 64.Among the known results of Noether's problem for non-abelian p-groups, except the situations in Theorem 1.1, assumptions on the existence of "enough" roots of unity always arose (see, for example, the following Theorem 1.3). In fact, even when G is a non-abelian p-group of order p 3 where p is an odd prime number, it is not known how to find a necessary and sufficient condition such as É(G) is rational over É (see [Ka5]). Similarly, without assuming the existence of roots of unity, we don't have a good criterion to guarantee É(G) is rational where G is a non-abelian group of order 32 (compare with the resu...

In this article we survey and examine the realizability of p-groups as Galois groups over arbitrary fields. In particular we consider various cohomological criteria that lead to necessary and sufficient conditions for the realizability of such a group as a Galois group, the embedding problem (i.e., realizability over a given subextension), descriptions of such extensions, automatic realizations among p-groups, and related topics.

In loving memory of my dear mother Let K be a field and G a finite group. Let G act on the rational function field K (x(g) : g ∈ G) by K-automorphisms defined by g • x(h) = x(gh) for any g, h ∈ G. Denote by K (G) the fixed field K (x(g) : g ∈ G) G. Noether's problem then asks whether K (G) is rational (i.e., purely transcendental) over K. The first main result of this article is that K (G) is rational over K for a certain class of p-groups having an abelian subgroup of index p. The second main result is that K (G) is rational over K for any group of order p 5 or p 6 (where p is an odd prime) having an abelian normal subgroup such that its quotient group is cyclic. (In both theorems we assume that if char K = p then K contains a primitive p e-th root of unity, where p e is the exponent of G.

In [Michailov I.M., On Galois cohomology and realizability of 2-groups as Galois groups, Cent. Eur. J. Math., 2011, 9(2), 403–419] we calculated the obstructions to the realizability as Galois groups of 14 non-abelian groups of order 2n, n ≥ 4, having a cyclic subgroup of order 2n−2, over fields containing a primitive 2n−3th root of unity. In the present paper we obtain necessary and sufficient conditions for the realizability of the remaining 8 groups that are not direct products of smaller groups.