On theta pairs for a maximal subgroup
HTML articles powered by AMS MathViewer
- by N. P. Mukherjee and Prabir Bhattacharya PDF
- Proc. Amer. Math. Soc. 109 (1990), 589-596 Request permission
Abstract:
For a maximal subgroup $M$ of a finite group $G$, a $\Theta$-pair is any pair of subgroups $(C,D)$ of $G$ such that (i) $D \triangleleft G,D \subset C$, (ii) $\left \langle {M,C} \right \rangle = G,\left \langle {M,D} \right \rangle = M$ and (iii) $C/D$ has no proper normal subgroup of $G/D$. A natural partial ordering is defined on the family of $\Theta$-pairs. We obtain several results on the maximal $\Theta$-pairs which imply $G$ to be solvable, supersolvable, and nilpotent.References
- Reinhold Baer, Classes of finite groups and their properties, Illinois J. Math. 1 (1957), 115–187. MR 87655 H. C. Bhatia, A generalized Frattini subgroup of a finite group, Ph. D. thesis, Michigan State Univ., East Lansing, MI, 1972.
- Prabir Bhattacharya and N. P. Mukherjee, On the intersection of a class of maximal subgroups of a finite group. II, J. Pure Appl. Algebra 42 (1986), no. 2, 117–124. MR 857561, DOI 10.1016/0022-4049(86)90074-5
- W. E. Deskins, On maximal subgroups, Proc. Sympos. Pure Math., Vol. 1, American Mathematical Society, Providence, R.I., 1959, pp. 100–104. MR 0125157
- Xiao Jing Wang, Pin Chao Wang, Jie Qing Lu, and Qiu Xiang Liu, The index complex of a maximal subgroup, Qufu Shifan Daxue Xuebao Ziran Kexue Ban 24 (1998), no. 3, 5–8 (Chinese, with English and Chinese summaries). MR 1665102
- Bertram Huppert, Normalteiler und maximale Untergruppen endlicher Gruppen, Math. Z. 60 (1954), 409–434 (German). MR 64771, DOI 10.1007/BF01187387
- B. Huppert, Endliche Gruppen. I, Die Grundlehren der mathematischen Wissenschaften, Band 134, Springer-Verlag, Berlin-New York, 1967 (German). MR 0224703
- N. P. Mukherjee, The hyperquasicenter of a finite group. I, Proc. Amer. Math. Soc. 26 (1970), 239–243. MR 268267, DOI 10.1090/S0002-9939-1970-0268267-1
- N. P. Mukherjee and Prabir Bhattacharya, The normal index of a maximal subgroup of a finite group, Proc. Amer. Math. Soc. 106 (1989), no. 1, 25–32. MR 952319, DOI 10.1090/S0002-9939-1989-0952319-9
- Henry G. Bray, W. E. Deskins, David Johnson, John F. Humphreys, B. M. Puttaswamaiah, Paul Venzke, and Gary L. Walls, Between nilpotent and solvable, Polygonal Publ. House, Washington, N.J., 1982. Edited and with a preface by Michael Weinstein. MR 655785
Additional Information
- © Copyright 1990 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 109 (1990), 589-596
- MSC: Primary 20D10; Secondary 20D25
- DOI: https://doi.org/10.1090/S0002-9939-1990-1015683-9
- MathSciNet review: 1015683