Crestani, E., Lucchini, A. (2015). Bias of group generators in finite and profinite groups: known results and open problems. International Journal of Group Theory, 4(2), 49-67. doi: 10.22108/ijgt.2015.9895

Eleonora Crestani; Andrea Lucchini. "Bias of group generators in finite and profinite groups: known results and open problems". International Journal of Group Theory, 4, 2, 2015, 49-67. doi: 10.22108/ijgt.2015.9895

Crestani, E., Lucchini, A. (2015). 'Bias of group generators in finite and profinite groups: known results and open problems', International Journal of Group Theory, 4(2), pp. 49-67. doi: 10.22108/ijgt.2015.9895

Crestani, E., Lucchini, A. Bias of group generators in finite and profinite groups: known results and open problems. International Journal of Group Theory, 2015; 4(2): 49-67. doi: 10.22108/ijgt.2015.9895

Bias of group generators in finite and profinite groups: known results and open problems

^{}Dipartimento di Matematica Universita; di Padova

Abstract

We analyze some properties of the distribution $Q_{G,k}$ of the first component in a $k$-tuple chosen uniformly in the set of all the $k$-tuples generating a finite group $G$ (the limiting distribution of the product replacement algorithm). In particular, we concentrate our attention on the study of the variation distance $\beta_k(G)$ between $Q_{G,k}$ and the uniform distribution. We review some known results, analyze several examples and propose some intriguing open questions.

L. Babai and I. Pak (2000). Strong bias of group generators: an obstacle to the "product replacement algorithm". Pro ceed-ings of the Eleventh Annual ACM-SIAM Symp osium on Discrete Algorithms (San Francisco, CA, 2000), ACM, New
York. , 627-635

2

F. Celler, C. R. Leedham-Green, S. Murray, A. Niemeyer and E. A. OBrien (1995). Generating random elements of a finite group. Comm. Algebra. 23, 4931-4948

3

E. Crestani and A. Lucchini Bias of group generators in the solvable case. Israel J. Math., to app ear, DOI:
10.1007/s11856-015-1159-7.

4

E. Crestani, G. De Franceschi and A. Lucchini Probability and bias in generating supersoluble groups. Proc. Edinb. Math. Soc , to appear.

5

J. D. Dixon (1969). The probability of generating the symmetric group. Math. Z.. 110, 199-205

6

M. D. Fried and M. Jarden (1986). Field Arithmetic. Ergebnisse der Mathematik und ihrer Grenzgebiete, Springer-Verlag, New York. 11

7

P. Hall (1936). The Eulerian functions of a group. Quart. J. Math.. 7, 134-151

8

W. M. Kantor and A. Lubotzky (1990). The probability of generating a finite classical group. Geom. Dedicata. 36 (1), 67-87

9

M. W. Liebeck and A. Shalev (1995). The probability of generating a finite simple group. Geom. Dedicata. 56 (1), 103-113

10

A. Lucchini (2005). The X-Dirichlet polynomial of a finite group. J. Group Theory. 8 (2), 171-188

11

A. Lucchini, F. Menegazzo and M. Morigi (2006). On the probability of generating prosoluble groups. Israel J. Math.. 155, 93-115

12

A. Lubotzky and I. Pak (2001). The pro duct replacement algorithm and Kazhdan's
property (T). J. Amer. Math. Soc.. 14 (2), 347-363

13

M. Morigi (2006). On the probability of generating free prosoluble groups of small rank. Israel J. Math.. 155, 177-123

14

A. Mann (1996). Positively finitely generated groups. Forum. Math.. 8 (4), 429-459

15

A. Mann and A. Shalev (1996). Simple groups, maximal subgroups, and probabilistic
aspects of profinite groups. Israel J. Math.. 96 part B, 449-468

16

N. E. Menezes, M. Quick and C. M. Roney-Dougal (2013). The probability of generating a nite simple group. Israel J. Math.. 198 (1), 371-392

17

I. Pak (2001). What do we know about the product replacement algorithm. in Groups and computation, I I I, de Gruyter, Berlin. , 301-347

18

M. Pinter (2010). The existence of an inverse limit of an inverse system of measure spaces - a purely measurable case. Acta Math. Hungar.. 126 (1-2), 65-77