###### Prof. Kaplan Gil

###### Fields of Research

###### School of Computer Science

Permutation Groups

Special subgroups like the Fitting, Frattini and commutator subgroups

Solvability and nilpotency conditions for finite groups

Aspects of maximal subgroups of finite groups

Products of subgroups in finite groups

## Short Bio

Gil completed his BSc in Mathematics and Physics in the Hebrew university in Jerusalem; his MSc and PhD in Mathematics were received in Tel Aviv University; the latter, under the supervision of Prof. Marcel Herzog, was focused on a problem in permutation groups. He published till now (2020) over 40 papers, many of them in leading journals like Journal of Group Theory, Journal of Algebra and Archiv der Mathematik. Gil was a regular guest in Bielefeld University, Germany, Vienna University of Technology, Berlin Free University, Würzburg University, Germany and Valencia University, Spain, where he gave many colloquium talks on his work. Gil supervised till now one student in the second degree programme of the Academic College Tel Aviv Yaffo; the work supervised was focused on regular decompositions of graphs.

Gil's areas of interest include Group Theory, Combinatory and some philosophical subjects.

## Selected Publications

1. H. Meng, A. Ballester-Bolinches and G. Kaplan, A note on maximal subgroups and conjugacy classes of finite groups, Quaestiones Mathematicae 45 (2022), 689-694.

2. M. Herzog, G. Kaplan, A. Lev and R. Zigdon, r-critical numbers of natural intervals, Electronic Journal of Combinatorics 28 (2021).

3. G. Kaplan, A note on Frobenius-Wielandt groups, Journal of Group Theory 22 (2019), 637-645.

4. B. Baumeister and G. Kaplan, Critical maximal subgroups and conjugacy of supplements in finite soluble groups, Journal of Group Theory 21 (2018**), **45-63.

5. G. Kaplan, A. Lev, A. Mann and R. Salomon, Adequate field extensions and Frattini closed groups, Communications in Algebra 45 (2017), 4622-4630.

6. B. Baumeister, G. Kaplan and D. Levy, Covering a finite group by the conjugates of a coset, Journal of Algebra 448 (2016), 84-103.

7. G. Kaplan and G. Vincenzi, On the Wielandt subgroup of generalized FC- groups, International Journal of Algebra and Computation 24 (2014), 1031-1042.

8. M. Herzog, G. Kaplan, P. Longobardi and M. Maj, Products of subsets of groups by their inverses, Beitraege zur Algebra und Geometrie 55 (2014), 311-346.

9. H. Heineken, W. Herfort and G. Kaplan, Nilpotency, solvability and the twisting function of finite groups II, Archiv der Mathematik 102 (2014), 501-512.

10. M. Herzog, G. Kaplan and A. Lev, Critical numbers of natural intervals, Journal of Number Theory 138 (2014), 69-83.

11. B. Baumeister and G. Kaplan, c-Sections, solvability and large subgroups of finite groups, Bulletin of the Australian Mathematical Society 86 (2012), 291-302.

12. G. Kaplan, On finite T-groups and the Wielandt subgroup, Journal of Group Theory 14 (2011), 855-863

13. G. Kaplan, On T-groups, supersolvable groups and maximal subgroups, Archiv der Mathematik 96 (2011), 19-25.

14. G. Kaplan, Nilpotency, solvability and the twisting function of finite groups, Communications in Algebra 39 (2011), 1722-1729.

15. G. Kaplan and D. Levy, Solvability of finite groups via conditions on products of 2-elements and odd p-elements, Bulletin of the Australian Mathematical Society 82 (2010), 265-273.

16. G. Kaplan and D. Levy, Solitary subgroups, Communications in Algebra 37 (2009), 1873-1883.

17. G. Kaplan, A. Lev and Y. Roditty, On zero-sum partitions and anti-magic trees, Discrete Mathematics 309 (2009), 2010-2014.

18. G. Kaplan, A. Lev and Y. Roditty, On graph labeling problems and regular decompositions of complete graphs, Ars Combinatoria 92 (2009), 225-243.

19. G. Kaplan and D. Levy, Multiplicities in Sylow sequences and the solvable radical, Bulletin of the Australian Mathematical Society 78 (2008), 477-486.

20. G. Kaplan and D. Levy, Schreier conditions on chief factors and residuals of solvable like group formations, Bulletin of the Australian Mathematical Society 78 (2008), 97-106.

21. G. Kaplan, A. Lev and Y. Roditty, Bertrand's postulate, the prime number theorem and product anti-magic graphs, Discrete Mathematics 308 (2008), 787-794.

22. M. Herzog, G. Kaplan and A. Lev, Covering the alternating groups by products of cycle classes, Journal of Combinatorial Theorey Ser. A 115 (2008), 1235-1245.

23. G. Kaplan and D. Levy, Sylow products and the solvable residual, Communications in Algebra 36 (2008), 851-857.

24. M. Herzog, G. Kaplan and A. Lev, The size of the commutator subgroup of finite groups, Journal of Algebra 320 (2008), 980-986.

25. S. Dolfi, M. Herzog, G. Kaplan and A. Lev, The size of the solvable residual in finite groups, Groups, Geometry and Dynamics 1 (2007), 401-407.

26. S. Dolfi, M. Herzog, G. Kaplan and A. Lev, On the size of the nilpotent residual in finite groups, Journal of Group Theory 10 (2007), 299-305.

27. G. Kaplan and D. Levy, Products of subgroups which are subgroups, Communications in Algebra 35 (5) (2007), 1517-1523.

28. M. Herzog, G. Kaplan and A. Lev, On the size of the commutator subgroup in finite groups, Groups St. Andrews 2005 in St. Andrews, Lecture Note Series, 486-490 (survey article).

29. G. Kaplan and D. Levy, Products of Sylow subgroups and the solvable radical, Groups St. Andrews 2005 in St. Andrews, Lecture Note Series, 527-530 (survey article).

30. G. Kaplan and D. Levy, Uniquely factorizable elements and solvability of finite groups, Archiv der Mathematik 86 (2006), 490-498.

31. G. Kaplan and D. Levy, The solvable radical of Sylow factorizable groups, Archiv der Mathematik 85 (2005), 490-496.

32. G. Kaplan and D. Levy, Sylow products and the solvable radical, Archiv der Mathematik 85 (2005), 304-312.

33. G. Kaplan, The existence of normal and characteristic subgroups in finite groups, Contemporary Mathematics, Ischia Group Theory 2004 (Proceedings of a conference in honor of Marcel Herzog, AMS), 197-200 (survey article, following an invited talk in Ischia Group Theory 2004).

34. G. Kaplan and A. Lev, The existence of large commutator subgroups in factors and subgroups of non-nilpotent groups, Archiv der Mathematik 85 (2005), 197-202.

35. G. Kaplan and D. Levy, GCD of truncated rows in Pascal triangle, Integers: The Electronic Journal of Combinatorial Number Theory 4 (2004).

36. N. Alon, G. Kaplan, A. Lev, Y. Roditty and R. Yuster, Dense graphs are antimagic, Journal of Graph Theory 47 (4) (2004), 297-309.

37. M. Herzog, G. Kaplan and A. Lev, Representation of permutations as products of two cycles, Discrete Mathematics 285 (2004), 323-327.

38. M. Herzog, G. Kaplan and A. Lev, On the commutator and the center of finite groups, Journal of Algebra 278 (2004), 494-501.

39. M. Herzog, G. Kaplan and A. Lucchini, On subgroups containing non-trivial normal subgroups, Israel Journal of Mathematics 137 (2003), 183-188.

40. G. Kaplan and A. Lev, On the dimension and basis concepts in finite groups, Communications in Algebra 31 (6) (2003), 2707-2717.

41. G. Kaplan and A. Lev, Covering numbers for non-perfect finite groups, Communications in Algebra 30 (11) (2002), 5253-5271.

42. G. Kaplan and A. Lev, Transversals in permutation groups and factorizations of complete graphs, Bulletin of the Australian Mathematical Society 65 (2002), 277-288.

43. G. Kaplan, A. Lev and Y. Roditty, Transitive large sets of disjoint decompositions and group sequencings, Discrete Mathematics 254 (2002), 207-219.

44. G. Kaplan, A. Lev and Y. Roditty, Gracefullness, group sequencings and graph factorizations, Groups St. Andrews 2001 in Oxford, Lecture Note Series, 295-305, Cambridge University Press (survey article).

45. M. Herzog and G. Kaplan, Large cyclic subgroups contain non-trivial normal subgroups, Journal of Group Theory 4 (2001), 247-253.

46. G. Kaplan, A. Lev and Y. Roditty, Regular Oberwolfach problems and group sequencings, Journal of Combinatorial Theory, Ser. A 96 (2001), 1-19.

47. G. Kaplan, Irreducible disconnected systems in groups, Israel Journal of Mathematics 111 (1999), 203-219.

48. G. Kaplan, On groups admitting a disconnected common divisor graph, Journal of Algebra 193 (1997), 616-628.

**Articles Accepted for Publication**

49. A. Ballester-Bolinches, G. Kaplan and V. Perez-Calabuig, On products of subgroups of finite groups, Mediterranean Journal of Mathematics, to appear.

**Active Participation in Scientific Meetings**

G. Kaplan, **Coprimeness and group structure (new and old results),**

Colloquium talk in the Algebra Seminar of the Vienna University of Technology, Austria, April 2016.

G. Kaplan, **Covering a finite group by the conjugates of a coset, **Colloquium talk in the Algebra Seminar of the University of Bielefeld, Germany, September 2015.

G. Kaplan, **New results on nilpotency, solvability and the twisting function of finite groups**, Colloquium talk in the Algebra Seminar of the University of Bielefeld, Germany, October 2014.

G. Kaplan, **Nilpotency, solvability and the twisting function of finite groups**, Colloquium talk in the Algebra Seminar of the University of Valencia, Spain**,**

May 2014.

G. Kaplan, **Some new results on finite T-groups**, Colloquium talk in the Algebra Seminar of the University of Bielefeld, Germany, October 2011.

G. Kaplan, **Nilpotency, solvability and the twisting function of finite groups**, Colloquium talk in the Algebra Seminar of the University of Bielefeld, Germany,

April 2011.

G. Kaplan**, Nilpotency, solvability and the twisting function of finite groups**, Colloquium talk in the Algebra Seminar of the Vienna University of Technology, Austria, September 2010.

G. Kaplan, **When must a subgroup contain a non-trivial normal subgroup?**

Colloquium talk in the Mathematical Institute of the Free University of Berlin, Germany, July 2009 (during a sabbatical year).

G. Kaplan, **When must a subgroup contain a non-trivial normal subgroup?**,

Colloquium talk in the Mathematical Institute of the University of Tuebingen,

Germany, June 2009 (during a sabbatical year).

G. Kaplan, **On the centre and the commutator subgroup of finite groups**,

Colloquium talk in the Algebra Seminar in the Mathematical Institute of the Technion, Haifa, April 2009.

G. Kaplan, **When must a subgroup contain a non-trivial normal subgroup?**,

Colloquium talk in the Algebra Seminar of the Vienna University of Technology, Austria, April 2008.

G. Kaplan and D. Levy, **Sylow products and solvability**,

Colloquium talk in the Mathematical Institute of the Free University of Berlin, Germany, July 2007.

G. Kaplan, **On the centre and commutator subgroup of finite groups**,

Colloquium talk in the Algebra Seminar of the Vienna University of Technology, Austria, April 2007.

G. Kaplan, **On the size of the commutator subgroup in finite groups**,

Colloquium talk in the Mathematical Institute of the Free University of Berlin, Germany, July 2006.

G. Kaplan, **Products of Sylow subgroups and the solvable radical**,

Colloquium talk in the Algebra Seminar of the Vienna University of Technology, Austria, April 2006.

G. Kaplan and D. Levy, **Products of Sylow subgroups and the solvable radical**, Conference talk in Groups St. Andrews 2005, St. Andrews, Scotland, August 2005.

G. Kaplan, Sylow products, **Sylow factorizability and the solvable radical**,

Colloquium talk in the Mathematical Institute of the University of Tuebingen, Germany, April 2005.

G. Kaplan, **The existence of normal and characteristic subgroups in finite groups**,

Conference talk in Ischia Group Theory 2004 (in honour of Marcel Herzog),

Naples, Italy, April 2004.

G. Kaplan, **The existence of normal and characteristic subgroups in finite groups**,

Colloquium talk in the Mathematical Seminar of the University of Kiel, Germany,

July 2003.

G. Kaplan**, On groups with large cyclic subgroups**, Colloquium talk in the Mathematical Institute of the University of Wuerzburg, Germany, June 2002.

G. Kaplan, **On groups admitting a disconnected common divisor graph**,

Colloquium talk in the Mathematical Institute of the University of Tuebingen, Germany, June 2001.

G. Kaplan**, On groups admitting a disconnected common divisor graph**,

Colloquium talk in the Mathematical Institute of the University of Wuerzburg, Germany, September 2000.