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GROUPS OF INFINITE RANK IN WHICH NORMALITY IS A TRANSITIVE RELATION

  • M. DE FALCO (a1), F. DE GIOVANNI (a1), C. MUSELLA (a1) and Y. P. SYSAK (a2)

Abstract

A group is called a T-group if all its subnormal subgroups are normal. It is proved here that if G is a periodic (generalized) soluble group in which all subnormal subgroups of infinite rank are normal, then either G is a T-group or it has finite rank. It follows that if G is an arbitrary group whose Fitting subgroup has infinite rank, then G has the property T if and only if all its subnormal subgroups of infinite rank are normal.

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References

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1.De Falco, M., de Giovanni, F., Musella, C. and Sysak, Y. P., On metahamiltonian groups of infinite rank, J. Algebra (to appear).
2.De Falco, M., de Giovanni, F., Musella, C. and Trabelsi, N., Groups whose proper subgroups of infinite rank have finite conjugacy classes, Bull. Austral. Math. Soc. (to appear).
3.De Mari, F. and de Giovanni, F., Groups satisfying the maximal condition on subnormal non-normal subgroups, Colloquium Math. 103 (2005), 8598.
4.De Mari, F. and de Giovanni, F., Groups satisfying the minimal condition on subnormal non-normal subgroups, Algebra Colloq. 13 (2006), 411420.
5.Dixon, M. R., Evans, M. J. and Smith, H., Locally soluble-by-finite groups of finite rank, J. Algebra 182 (1996), 756769.
6.Dixon, M. R., Evans, M. J. and Smith, H., Locally (soluble-by-finite) groups with all proper non-nilpotent subgroups of finite rank, J. Pure Appl. Algebra 135 (1999), 3343.
7.Dixon, M. R., Evans, M. J. and Smith, H., Groups with all proper subgroups (finite rank)-by-nilpotent, Arch. Math. (Basel) 72 (1999), 321327.
8.Dixon, M. R. and Karatas, Z. Y., Groups with all subgroups permutable or of finite rank, Centr. Eur. J. Math. 10 (2012), 950957.
9.Evans, M. J. and Kim, Y., On groups in which every subgroup of infinite rank is subnormal of bounded defect, Comm. Algebra 32 (2004), 25472557.
10.Franciosi, S. and de Giovanni, F., Groups in which every infinite subnormal subgroup is normal, J. Algebra 96 (1985), 566580.
11.Gaschütz, W., Gruppen in denen das Normalteilersein transitiv ist, J. Reine Angew. Math. 198 (1957), 8792.
12.Robinson, D. J. S., Groups in which normality is a transitive relation, Proc. Camb. Philos. Soc. 68 (1964), 2138.
13.Robinson, D. J. S., Finiteness conditions and generalized soluble groups (Springer, Berlin, Germany, 1972).
14.Semko, N. N. and Kuchmenko, S. N.: Groups with almost normal subgroups of infinite rank, Ukrain. Math. J. 57 (2005), 621639.

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GROUPS OF INFINITE RANK IN WHICH NORMALITY IS A TRANSITIVE RELATION

  • M. DE FALCO (a1), F. DE GIOVANNI (a1), C. MUSELLA (a1) and Y. P. SYSAK (a2)

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