Structure of Groups with Generalized Normal Subgroups

ABSTRACT

The
present paper deals with a subgroup *X* of a group *G* is almost normal if the
index |*G*: *NG*(*X*)| is finite, while *X* is nearly normal if it has finite
index in the normal closure *XG*. This
paper investigates the structure of groups in which every (infinite) subgroup
is either almost normal or nearly normal.

Cite this paper

Arshaduzzaman, M. (2014) Structure of Groups with Generalized Normal Subgroups.*Advances in Pure Mathematics*, **4**, 261-268. doi: 10.4236/apm.2014.46035.

Arshaduzzaman, M. (2014) Structure of Groups with Generalized Normal Subgroups.

References

[1] Neumann, B.H. (1955) Groups with Finite Classes of Conjugate Subgroups. Mathematische Zeitschrift, 63, 76-96.

http://dx.doi.org/10.1007/BF01187925

[2] Eremin, I.I. (1959) Groups with Finite Classes of Conjugate Abelian Subgroups. Matematicheskii Sbornik (MatSb) Moscow, 47, 45-54.

[3] Tomkinson, M.J. (1981) On Theorems of B. H. Neumann Concerning F C-Groups. Rocky Mountain Journal of Mathematics, 11, 47-58.

http://dx.doi.org/10.1216/RMJ-1981-11-1-47

[4] Robinson, D.J.S. (1972) Finiteness Conditions and Generalized Soluble Groups. Springer, Berlin.

[5] Kurdachenko, L.A., Levishchenko, S.S. and Semko, N.N. (1983) Groups with Almost Normal Infinite Subgroups. Soviet Mathematics (Izvestiya VUZ. Matematika), 27, 73-81.

[6] Franciosi, S. and de Giovanni, F. (1994 & 1995) Groups Satisfying the Minimal Condition on Certain Non-Normal Subgroups. Proceedings of Groups—Korea, de Gruyter, Berlin, 107-118.

[7] Polovickii, Y.D. (1980) Groups with Finite Classes of Conjugate Infinite Abelian Subgroups. Soviet Mathematics (Izvestiya VUZ. Matematika), 24, 52-59.

[8] De Falco, M., de Giovanni, F. and Musella, C. (2010) Groups with Finitely Many Normalizers of Non-Polycyclic Subgroups. Algebra Colloquium (World Scientific), 17, 203-210.

http://dx.doi.org/10.1142/S1005386710000210

[9] De Mari, F. and de Giovanni, F. (2005) Groups with Few Normalizer Subgroups. Irish Mathematical Society Bulletin, 56, 103-113.

[10] De Mari, F. and de Giovanni, F. (2006) Groups with Finitely Many Normalizers of Non-Abelian Subgroups. Ricerche di Matematica, 55, 311-317.

[11] De Mari, F. and de Giovanni, F. (2006) Groups with Finitely Many Normalizers of Subnormal Subgroups. Journal of Algebra, 304, 382-396.

http://dx.doi.org/10.1016/j.jalgebra.2005.10.006

[12] De Mari, F. and de Giovanni, F. (2007) Groups with Finitely Many Normalizers of Non-Nilpotent Subgroups. Mathematical Proceedings of the Royal Irish Academy. Royal Irish Academy, Dublin, 107A, 143-152.

http://dx.doi.org/10.3318/PRIA.2007.107.2.143

[13] Tomkinson, M.J. (1984) FC-Groups, Pitman, Boston.

[14] Neumann, B.H. (1954) Groups Covered by Permutable Subset. Journal of the London Mathematical Society, 29, 236-248.

http://dx.doi.org/10.1112/jlms/s1-29.2.236

[15] Mann, A. and Segal, D. (1990) Uniform Finiteness Conditions in Residually Finite Groups. Proceedings London Mathematical Society, 61, 529-545.

[16] Sunkov, V.P. (1970) On the Minimality Problem for Locally Finite Groups. Algebra and Logic, 9, 137-151.

http://dx.doi.org/10.1007/BF02218982

[17] Hartley, B. (1988) Fixed Points of Automorphisms of Certain Locally Finite Groups and Chevalley Groups. Journal of the London Mathematical Society, 37, 421-436.

http://dx.doi.org/10.1112/jlms/s2-37.3.421

[18] Zaicev, D.I. (1974) On Solvable Subgroups of Locally Solvable Groups. Soviet Mathematics Doklady (SSSR), 15, 342-345.

[1] Neumann, B.H. (1955) Groups with Finite Classes of Conjugate Subgroups. Mathematische Zeitschrift, 63, 76-96.

http://dx.doi.org/10.1007/BF01187925

[2] Eremin, I.I. (1959) Groups with Finite Classes of Conjugate Abelian Subgroups. Matematicheskii Sbornik (MatSb) Moscow, 47, 45-54.

[3] Tomkinson, M.J. (1981) On Theorems of B. H. Neumann Concerning F C-Groups. Rocky Mountain Journal of Mathematics, 11, 47-58.

http://dx.doi.org/10.1216/RMJ-1981-11-1-47

[4] Robinson, D.J.S. (1972) Finiteness Conditions and Generalized Soluble Groups. Springer, Berlin.

[5] Kurdachenko, L.A., Levishchenko, S.S. and Semko, N.N. (1983) Groups with Almost Normal Infinite Subgroups. Soviet Mathematics (Izvestiya VUZ. Matematika), 27, 73-81.

[6] Franciosi, S. and de Giovanni, F. (1994 & 1995) Groups Satisfying the Minimal Condition on Certain Non-Normal Subgroups. Proceedings of Groups—Korea, de Gruyter, Berlin, 107-118.

[7] Polovickii, Y.D. (1980) Groups with Finite Classes of Conjugate Infinite Abelian Subgroups. Soviet Mathematics (Izvestiya VUZ. Matematika), 24, 52-59.

[8] De Falco, M., de Giovanni, F. and Musella, C. (2010) Groups with Finitely Many Normalizers of Non-Polycyclic Subgroups. Algebra Colloquium (World Scientific), 17, 203-210.

http://dx.doi.org/10.1142/S1005386710000210

[9] De Mari, F. and de Giovanni, F. (2005) Groups with Few Normalizer Subgroups. Irish Mathematical Society Bulletin, 56, 103-113.

[10] De Mari, F. and de Giovanni, F. (2006) Groups with Finitely Many Normalizers of Non-Abelian Subgroups. Ricerche di Matematica, 55, 311-317.

[11] De Mari, F. and de Giovanni, F. (2006) Groups with Finitely Many Normalizers of Subnormal Subgroups. Journal of Algebra, 304, 382-396.

http://dx.doi.org/10.1016/j.jalgebra.2005.10.006

[12] De Mari, F. and de Giovanni, F. (2007) Groups with Finitely Many Normalizers of Non-Nilpotent Subgroups. Mathematical Proceedings of the Royal Irish Academy. Royal Irish Academy, Dublin, 107A, 143-152.

http://dx.doi.org/10.3318/PRIA.2007.107.2.143

[13] Tomkinson, M.J. (1984) FC-Groups, Pitman, Boston.

[14] Neumann, B.H. (1954) Groups Covered by Permutable Subset. Journal of the London Mathematical Society, 29, 236-248.

http://dx.doi.org/10.1112/jlms/s1-29.2.236

[15] Mann, A. and Segal, D. (1990) Uniform Finiteness Conditions in Residually Finite Groups. Proceedings London Mathematical Society, 61, 529-545.

[16] Sunkov, V.P. (1970) On the Minimality Problem for Locally Finite Groups. Algebra and Logic, 9, 137-151.

http://dx.doi.org/10.1007/BF02218982

[17] Hartley, B. (1988) Fixed Points of Automorphisms of Certain Locally Finite Groups and Chevalley Groups. Journal of the London Mathematical Society, 37, 421-436.

http://dx.doi.org/10.1112/jlms/s2-37.3.421

[18] Zaicev, D.I. (1974) On Solvable Subgroups of Locally Solvable Groups. Soviet Mathematics Doklady (SSSR), 15, 342-345.