Largeness and equational probability in groups
Annales mathématiques Blaise Pascal, Tome 27 (2020) no. 1, pp. 1-17.

We define k-genericity and k-largeness for a subset of a group, and determine the value of k for which a k-large subset of G n is already the whole of G n , for various equationally defined subsets. We link this with the inner measure of the set of solutions of an equation in a group, leading to new results and/or proofs in equational probabilistic group theory.

Publié le :
DOI : 10.5802/ambp.388
Classification : 20A15, 03C60, 20P99
Mots clés : probabilistic group theory, largeness
Khaled Jaber 1 ; Frank O. Wagner 2

1 Department of Mathematics Faculty of Sciences Lebanese University, Lebanon
2 Université de Lyon; Université Lyon 1 CNRS UMR 5208, Institut Camille Jordan 21 avenue Claude Bernard 69622 Villeurbanne-cedex, France
Licence : CC-BY 4.0
Droits d'auteur : Les auteurs conservent leurs droits
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Khaled Jaber; Frank O. Wagner. Largeness and equational probability in groups. Annales mathématiques Blaise Pascal, Tome 27 (2020) no. 1, pp. 1-17. doi : 10.5802/ambp.388. https://ambp.centre-mersenne.org/articles/10.5802/ambp.388/

[1] Fran Barry; Des MacHale; Á. Ní Shé Some supersolvability conditions for finite groups, Math. Proc. R. Ir. Acad., Volume 106 (2006) no. 2, pp. 163-177 | DOI | MR | Zbl

[2] Roger Bryant Groups with the minimal condition on centralizers, J. Algebra, Volume 60 (1979), pp. 371-383 | DOI | MR

[3] Paul Erdős; Pál Turan On some problems of a statistical group theory. IV, Acta Math. Acad. Sci. Hung., Volume 19 (1968), pp. 413-435 | DOI | MR | Zbl

[4] D. G. Farrokhi On the probability that a group satisfies a law: A survey, Kyoto Univ. Res. Inform. Repos., Volume 1965 (2015), pp. 158-179

[5] Georg Frobenius Verallgemeinerung des Sylowschen Satzes, Sitzungsber. K. Preuss. Akad. Wiss. Berlin, Volume II (1895), pp. 981-993

[6] William H. Gustafson What is the probability that two group elements commute?, Am. Math. Mon., Volume 80 (1973), pp. 1031-1034 | DOI | MR | Zbl

[7] Robert Heffernan; Des MacHale; Á. Ní Shé Restrictions on commutativity ratios in finite groups, Int. J. Group Theory, Volume 3 (2014) no. 4, pp. 1-12 | MR

[8] Nobuo Iiyori; Hiroyoshi Yamaki On a conjecture of Frobenius, Bull. Am. Math. Soc., Volume 25 (1991) no. 2, pp. 413-416 | DOI | MR | Zbl

[9] Khaled Jaber; Frank O. Wagner Largeur et nilpotence, Commun. Algebra, Volume 28 (2000) no. 6, pp. 2869-2885 | DOI | MR | Zbl

[10] Keith S. Joseph Several conjectures on commutativity in algebraic structures, Am. Math. Mon., Volume 84 (1977), pp. 550-551 | DOI | MR

[11] Thomas J. Laffey The number of solutions of x 3 =1 in a 3-group, Math. Z., Volume 149 (1976), pp. 43-45 | DOI | MR | Zbl

[12] Thomas J. Laffey The number of solutions of x p =1 in a finite group, Proc. Camb. Philos. Soc., Volume 80 (1976), pp. 229-231 | DOI | MR | Zbl

[13] Thomas J. Laffey The number of solutions of x 4 =1 in finite groups, Math. Proc. R. Ir. Acad., Volume 79 (1979), pp. 29-36 | MR | Zbl

[14] George A. Miller Note on the possible number of operators of order 2 in a group of order 2 m , Ann. Math., Volume 7 (1906), pp. 55-60 | DOI | MR | Zbl

[15] Bernhard H. Neumann Groups covered by permutable subsets, J. Lond. Math. Soc., Volume 29 (1954), pp. 236-248 | DOI | MR

[16] Peter M. Neumann Two combinatorial problems in group theory, Bull. Lond. Math. Soc., Volume 21 (1989) no. 5, pp. 456-458 | DOI | MR | Zbl

[17] Mohammad R. Pournaki; Reza Sobhani Probability that the commutator of two group elements is equal to a given element, J. Pure Appl. Algebra, Volume 212 (2008) no. 4, pp. 727-734 | DOI | MR | Zbl

[18] David J. Rusin What is the probability that two elements of a finite group commute?, Pac. J. Math., Volume 82 (1979), pp. 237-247 | DOI | MR | Zbl

[19] Gary J. Sherman What is the probability an automorphism fixes a group element?, Am. Math. Mon., Volume 82 (1975), pp. 261-264 | DOI | MR | Zbl

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