On some metabelian 2-group and applications II
DOI:
https://doi.org/10.5269/bspm.v34i2.27016Abstract
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.
References
1. 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.
2. 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.
3. 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).
4. 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.
5. A. Azizi, A. Zekhnini and M. Taous, On some metabelian 2-group and applications I, to appear in Colloquium Mathematicum.
6. 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.
7. E. Benjamin and C. Snyder, Number Fields with 2-class Number Isomorphic to (2, 2m), preprint, 1994.
8. 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).
9. E. Benjamin, F. Lemmermeyer, C. Snyder, Imaginary quadratic fields with Cl2(k) . (2, 2, 2), J. Number Theory 103 (2003), 38-70.
10. F. Lemmermeyer, On 2-class field towers of some imaginary quadratic number fields, Abh. Math. Sem. Hamburg 67 (1997), 205-214
11. H. Kisilevsky, Number fields with class number ongruent to 4 mod 8 and Hilbert’s thorem 94, J. Number Theory 8 (1976), 271-279.
12. D. J. Robinson, A course in the Theory of Groups, 2nd ed. Springer-Verlag New York, (1996).
13. I. M. Isaacs, Character Theory of Finite Groups, New York: Academic Press, (1976).
14. K. Miyake, Algebraic Investigations of Hilbert’s Theorem 94, the Principal Ideal theorem and Capitulation Problem, Expos. Math. 7 (1989), 289-346.
15. P. Kaplan, Sur le 2-groupe de classes d’idéaux des corps quadratiques. J. Reine angew. Math. 283/284 (1976), 313-363.
2. 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.
3. 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).
4. 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.
5. A. Azizi, A. Zekhnini and M. Taous, On some metabelian 2-group and applications I, to appear in Colloquium Mathematicum.
6. 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.
7. E. Benjamin and C. Snyder, Number Fields with 2-class Number Isomorphic to (2, 2m), preprint, 1994.
8. 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).
9. E. Benjamin, F. Lemmermeyer, C. Snyder, Imaginary quadratic fields with Cl2(k) . (2, 2, 2), J. Number Theory 103 (2003), 38-70.
10. F. Lemmermeyer, On 2-class field towers of some imaginary quadratic number fields, Abh. Math. Sem. Hamburg 67 (1997), 205-214
11. H. Kisilevsky, Number fields with class number ongruent to 4 mod 8 and Hilbert’s thorem 94, J. Number Theory 8 (1976), 271-279.
12. D. J. Robinson, A course in the Theory of Groups, 2nd ed. Springer-Verlag New York, (1996).
13. I. M. Isaacs, Character Theory of Finite Groups, New York: Academic Press, (1976).
14. K. Miyake, Algebraic Investigations of Hilbert’s Theorem 94, the Principal Ideal theorem and Capitulation Problem, Expos. Math. 7 (1989), 289-346.
15. P. Kaplan, Sur le 2-groupe de classes d’idéaux des corps quadratiques. J. Reine angew. Math. 283/284 (1976), 313-363.
Downloads
Published
2015-06-29
Issue
Section
Research Articles
License
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).
