On some metabelian 2-group and applications II
Résumé
Let G be some metabelian 2-group satisfying the condition G/G' is of type (2, 2, 2). In this paper, we construct all the subgroups of G of index 2 or 4, we give the abelianization types of these subgroups and we compute the kernel of the transfer map. Then we apply these results to study the capitulation problem of the 2-ideal classes of some fields k satisfying the condition Gal(k_2^{(2)}/k) is isomorphic to G, where k_2^{(2)} is the second Hilbert 2-class field of k.
Téléchargements
Références
A. Azizi, A. Zekhnini and M. Taous, Coclass of Gal(k(2) 2 /k) for some fields k = Q..vp1p2q,v-1 with 2-class groups of type (2, 2, 2), to appear in J. Algebra Appl, (2015). DOI: 10.1142/S0219498816500274.
A. Azizi, A. Zekhnini and M. Taous, Structure of Gal(k(2) 2 /k) for some fields k = Q(v2p1p2, i) with Cl2(k) . (2, 2, 2), Abh. Math. Sem. Univ. Hamburg, Volume 84, 2 (2014), 203-231.
A. Azizi, A. Zekhnini, M. Taous and Daniel C. Mayer, Principalization of 2-class groups of type (2, 2, 2) of biquadratic fields Q(vp1p2q, i), Int. J. Number Theory, Vol. 11, No. 04, pp. 1177-1215 (2015).
A. Azizi, A. Zekhnini and M. Taous, On some metabelian 2-group whose abelianization is of type (2, 2, 2) and applications, J. Taibah Univ. Sci. (2015), http://dx.doi.org/10.1016/j.jtusci.2015.01.007.
A. Azizi, A. Zekhnini and M. Taous, On some metabelian 2-group and applications I, to appear in Colloquium Mathematicum.
A. Azizi, M. Taous and A. Zekhnini, On the 2-groups whose abelianizations are of type (2, 4) and applications, to appear in Publicationes Mathematicae Debrecen.
E. Benjamin and C. Snyder, Number Fields with 2-class Number Isomorphic to (2, 2m), preprint, 1994.
E. Benjamin, F. Lemmermeyer and C. Snyder, Real Quadratic Fields with Abelian 2-Class Field tower, J. of Number Theory, Volume 73, Number 2, December (1998), pp. 182-194 (13).
E. Benjamin, F. Lemmermeyer, C. Snyder, Imaginary quadratic fields with Cl2(k) . (2, 2, 2), J. Number Theory 103 (2003), 38-70.
F. Lemmermeyer, On 2-class field towers of some imaginary quadratic number fields, Abh. Math. Sem. Hamburg 67 (1997), 205-214
H. Kisilevsky, Number fields with class number ongruent to 4 mod 8 and Hilbert’s thorem 94, J. Number Theory 8 (1976), 271-279.
D. J. Robinson, A course in the Theory of Groups, 2nd ed. Springer-Verlag New York, (1996).
I. M. Isaacs, Character Theory of Finite Groups, New York: Academic Press, (1976).
K. Miyake, Algebraic Investigations of Hilbert’s Theorem 94, the Principal Ideal theorem and Capitulation Problem, Expos. Math. 7 (1989), 289-346.
P. Kaplan, Sur le 2-groupe de classes d’idéaux des corps quadratiques. J. Reine angew. Math. 283/284 (1976), 313-363.
When the manuscript is accepted for publication, the authors agree automatically to transfer the copyright to the (SPM).
The journal utilize the Creative Common Attribution (CC-BY 4.0).
