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COMPONENTS AND MINIMAL NORMAL SUBGROUPS OF FINITE AND PSEUDOFINITE GROUPS

  • JOHN S. WILSON (a1)

Abstract

It is proved that there is a formula $\pi \left( {h,x} \right)$ in the first-order language of group theory such that each component and each non-abelian minimal normal subgroup of a finite group G is definable by $\pi \left( {h,x} \right)$ for a suitable element h of G; in other words, each such subgroup has the form $\left\{ {x|x\pi \left( {h,x} \right)} \right\}$ for some h. A number of consequences for infinite models of the theory of finite groups are described.

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[1]Aschbacher, M. and Guralnick, R., Some applications of the first cohomology group. Journal of Algebra, vol. 90 (1984), pp. 446460.10.1016/0021-8693(84)90183-2
[2]Frink, O., Pseudo-complements in semi-lattices. Duke Mathematical Journal, vol. 29 (1962), pp. 505514.10.1215/S0012-7094-62-02951-4
[3]Felgner, U., Pseudo-endliche Gruppen, Proceedings of the 8th Easter Conference on Model Theory (Dahn, B. and Wolter, H., editors), Humboldt–Universität, Berlin, 1990, pp. 8296.
[4]Gorenstein, D., Finite Simple Groups, An Introduction to their Classification, Plenum Press, New York, 1982.10.1007/978-1-4684-8497-7
[5]Isaacs, I. M., Finite Group Theory, American Mathematical Society, Providence, RI, 2008.
[6]Liebeck, M. W., O’Brien, E. A., Shalev, A., and Tiep, P. H., The Ore conjecture. Journal of the European Mathematical Society, vol. 12 (2010), pp. 9391008.10.4171/JEMS/220
[7]Liebeck, M. W., O’Brien, E. A., Shalev, A., and Tiep, P. H., Commutators in finite quasisimple groups. Bulletin of the London Mathematical Society, vol. 43 (2011), pp. 10791092.10.1112/blms/bdr043
[8]Macpherson, D. and Tent, K., Stable pseudofinite groups. Journal of Algebra, vol. 312 (2007), pp. 550561.10.1016/j.jalgebra.2005.07.012
[9]Ould Houcine, A. and Point, F., Alternatives for pseudofinite groups. Journal of Group Theory, vol. 16 (2013), pp. 461495.10.1515/jgt-2013-0006
[10]Wilson, J. S., On simple pseudofinite groups. Journal of the London Mathematical Society (2), vol. 51 (1995), pp. 471490.10.1112/jlms/51.3.471
[11]Wilson, J. S., First-order characterization of the radical of a finite group, this Journal, vol. 74 (2009), pp. 14291435.
[12]Wilson, J. S., The first-order theory of branch groups. Journal of the Australian Mathematical Society, vol. 102 (2017), pp. 150158.

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