Powers in Alternating Simple Groups
DOI:
https://doi.org/10.17398/Keywords:
Alternating groups, simple groups, power subgroups, word mapsAbstract
C. Martínez and E. Zelmanov proved in [12] that for every natural number d and every finite simple group G, there exists a function N = N(d) such that either G^d = 1 or G = {a_1^d, ... , a_N^d : ai \in G}.
In a more general context the problem of finding words w such that the word map (g1; ...; gd) --> w(g1; ... ; gd) is surjective for any finite non abelian simple group is a major challenge in Group Theory. In [8] authors give the first example of a word map which is surjective on all finite non-abelian simple groups, the commutator [x; y] (Ore Conjecture).
In [11] the conjecture that this is also the case for the word x2y2 is formulated. This conjecture was solved in [9] and, independently, in [6], using deep results of algebraic simple groups and representation theory. An elementary proof of this result for alternating simple groups is presented here.
Downloads
References
E. Bertran, Even permutations as a product of two conjugate cycles, J. Combin. Theory Ser. A 12 (1972), 368 – 380.
E. Bertran, Powers of cycle-classes in symmetric groups, J. Combin. Theory Ser. A 94 (1) (2001), 87 – 99.
V.I. Chernousov, E.W. Ellers, N.L. Gordeev, Gauss decomposition with prescribed semisimple part: short proof, J. Algebra 229 (1) (2000), 314 – 332.
E.W. Ellers, N.L. Gordeev, On the conjectures of J. Thompson and O. Ore, Trans. Amer. Math. Soc. 350 (9) (1998), 3657 – 3671.
W.J. Ellision, Warings’s problem, Amer. Math. Monthly 78 (1) (1971), 10 – 36.
R. Guralnick, G. Malle, Products of conjugacy classes and fixed point spaces, J. Amer. Math. Soc. 25 (1) (2012), 77 – 121.
M. Larsen, A. Shalev, Word maps and Waring type problems, J. Amer. Math. Soc. 22 (2) (2009), 437 – 466.
M.W. Liebeck, E.A. O'Brien, A. Shalev, P.H. Tiep, The Ore conjecture, J. Eur. Math. Soc. (JEMS) 12 (4) (2010), 939 – 1008.
M.W. Liebeck, E.A. O'Brien, A. Shalev, P.H. Tiep, Products of squares in finite simple groups, Proc. Amer. Math. Soc. 140 (1) (2012), 21 – 33.
M.W. Liebeck, A. Shalev, Diameters of finite simple groups: sharp bounds and applications, Ann. of Math. (2) 154 (2) (2001), 383 – 406.
M.J. Larsen, A. Shalev, P.H. Tiep, The Waring problem for finite simple groups, Ann. of Math. (2) 174 (3) (2011), 1885 – 1950.
C. Martinez, E.I. Zelmanov, Products of powers in finite simple groups, Israel J. Math. 96 (1996), part B, 469 – 479.
O. Ore, Some remarks on commutators, Proc. Amer. Math. Soc. 2 (1952), 307 – 314.
J. Saxl, J.S. Wilson, A note on powers in simple groups, Math. Proc. Cambridge Philos. Soc. 122 (1) (1997), 91 – 94.
J.S. Wilson, First-order group theory, in “ Infinite groups 1994 ”, Proceedings of the International Conference held in Ravello, May 23-27, 1994 (edited by F. de Giovanni and M.L. Newell), Walter de Gruyter & Co., Berlin, 1996, 301 – 314.
