A note on finite groups with few TI-subgroups

Year 2018, Volume: 23 Issue: 23, 42 - 46, 11.01.2018

Jiangtao Shi Jingjing Huang Cui Zhang

https://doi.org/10.24330/ieja.373640

Cited By: 1

Abstract

In [Comm. Algebra, 43(2015), 2680-2689], nite groups all of

whose metacyclic subgroups are TI-subgroups have been classied by S. Li,

Z. Shen and N. Du. In this note we investigate a nite group all of whose

non-metacyclic subgroups are TI-subgroups. We prove that G is a group all

of whose non-metacyclic subgroups are TI-subgroups if and only if all non-

metacyclic subgroups of G are normal. Furthermore, we show that a group all

of whose non-cyclic subgroups are TI-subgroups has a Sylow tower.

Keywords

Non-metacyclic subgroup, TI-subgroup, normal, solvable, Sylow tower

References

• S. Li, Z. Shen and N. Du, Finite groups with few TI-subgroups, Comm. Algebra, 43(7) (2015), 2680-2689.
• H. Mousavi, T. Rastgoo and V. Zenkov, The structure of non-nilpotent CTI- groups, J. Group Theory, 16(2) (2013), 249-261.
• D. J. S. Robinson, A Course in the Theory of Groups, Second Edition, Grad- uate Texts in Mathematics, 80, Springer-Verlag, New York, 1996.
• J. Shi and C. Zhang, Finite groups in which some particular subgroups are TI-subgroups, Miskolc Math. Notes, 14(3) (2013), 1037-1040.