## Revista Matemática Iberoamericana

### On some permutable products of supersoluble groups

#### Abstract

It is well known that a group $G = AB$ which is the product of two supersoluble subgroups $A$ and $B$ is not supersoluble in general. Under suitable permutability conditions on $A$ and $B$, we show that for any minimal normal subgroup $N$ both $AN$ and $BN$ are supersoluble. We then exploit this to establish some sufficient conditions for $G$ to be supersoluble.

#### Article information

Source
Rev. Mat. Iberoamericana, Volume 20, Number 2 (2004), 413-425.

Dates
First available in Project Euclid: 17 June 2004

https://projecteuclid.org/euclid.rmi/1087482021

Mathematical Reviews number (MathSciNet)
MR2073126

Zentralblatt MATH identifier
1063.20024

#### Citation

Alejandre, Manuel J.; Ballester-Bolinches, A.; Cossey, John; Pedraza-Aguilera, M. C. On some permutable products of supersoluble groups. Rev. Mat. Iberoamericana 20 (2004), no. 2, 413--425. https://projecteuclid.org/euclid.rmi/1087482021

#### References

• Alejandre, Manuel J., Ballester-Bolinches, A., Cossey, John: Permutable products of supersoluble groups. To appear in J. Algebra.
• Amberg, B., Franciosi, S. and De Giovanni, F.: Products of groups. Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, New York, 1992.
• Asaad, M., Shaalan, A.: On the supersolubility of finite groups. Arch. Math. 53 (1989), no. 4, 318-326.
• Baer, R.: Classes of finite groups and their properties. Illinois J. Math. 1 (1957), 115-187.
• Ballester-Bolinches, A., Doerk, K. and Pérez-Ramos, M. D.: On the lattice of $\F$-subnormal groups. J. Algebra 148 (1992), no. 1, 42-52.
• Beidleman, J. C., Galoppo, A., Heineken, H. and Manfredino, M.: On certain products of soluble groups. Forum Math. 13 (2001), 569-580.
• Carocca, A.: $p$-supersolvability of factorized finite groups. Hokkaido Math. J. 21 (1992), 395-403.
• Carocca, A.: On factorized finite groups in which certain subgroups of the factors permute. Arch. Math. 71 (1998), 257-261.
• Doerk, K. and Hawkes, T. O.: Finite soluble groups. De Gruyter Expositions in Mathematics 4. Walter de Gruyter, Berlin, 1992.
• Förster, P.: Finite groups all whose subgroups are $\F$-subnormal or $\F$-subabnormal. J. Algebra 103 (1986), no. 1, 285-293.
• Kegel, O. H.: Sylow-Gruppen und Subnormalteiler endlicher Gruppen. Math. Z. 78 (1962), 205-221.
• Stonehewer, S. E.: On finite dinilpotent groups. J. Pure Appl. Algebra 88 (1993), no. 1-3, 239-244.