# On groups with two isomorphism classes of central factors

Document Type: Ischia Group Theory 2016

Author

Universitamp;agrave; degli Studi di Salerno

Abstract

The structure of groups which have at most two isomorphism classes of central factors ($B_2$-groups) are investigated‎. ‎A complete description of $B_2$-groups is obtained in the locally finite case and in the nilpotent case‎. ‎In addition detailed information is obtained about soluble $B_2$-groups‎. ‎Also structural information about insoluble $B_2$-groups is given‎, ‎in particular when such a group has the derived subgroup satisfying the minimal condition‎.

Keywords

Main Subjects

### References

[1] F. de Giovanni and D. J. S. Robinson, Groups with finitely many derived subgroups, J. London Math. Soc. (2), 71 (2005) 658–668.

[2] M. Herzog, P. Longobardi and M. Maj, On the number of commutators in groups, Ischia Group Theory 2004, Amer. Math. Soc., Providence, RI, 402 (2006) 181-192.

[3] J. C. Lennox, H. Smith and J. Wiegold, A problem about normal subgroups, J. Pure Appl. Algebra, 88 (1993) 169–171.

[4] P. Longobardi, M. Maj, D. J. S. Robinson and H. Smith, On groups with two isomorphism classes of derived
subgroups, Glasgow Math. J., 55 (2013) 655–668.

[5] P. Longobardi, M. Maj and D. J. S. Robinson, Recent results on groups with few isomorphism classes of derived subgroups, Proc. of ”Group Theory, Combinatorics, and Computing”, Boca Raton-Florida, Contemp. Math., 611 (2014) 121–135.

[6] P. Longobardi, M. Maj and D. J. S. Robinson, Locally finite groups with finitely many isomorphism classes of derived subgroups, J. Algebra, 393 (2013) 102–119.

[7] G. A. Miller and H. C. Moreno, Non-abelian groups in which every subgroup is abelian, Trans. Amer. Math. Soc., 4 (1903) 398-404.

[8] D. J. S. Robinson, A course in the theory of groups, Springer-Verlag, 1996.

[9] H. Smith, On homomorphic images of locally graded groups, Rend. Sem. Mat. Univ. Padova, 91 (1994) 53-60.

[10] H. Smith, J. Wiegold, Groups which are isomorphic to their non-abelian subgroups, Rend. Sem. Mat. Univ. Padova, 97 (1997) 7–16.