On the Number of Simple $$K_4$$ K 4 Groups
- PDF / 221,864 Bytes
- 6 Pages / 439.37 x 666.142 pts Page_size
- 44 Downloads / 188 Views
On the Number of Simple K4 Groups Shaohua Zhang1 · Wujie Shi2,3 Received: 30 July 2019 / Accepted: 30 December 2019 © The Author(s) 2020
Abstract In this paper, by solving Diophantine equations involving simple K 4 -groups, we will try to point out that it is not easy to prove the infinitude of simple K 4 -groups. This problem goes far beyond what is known about Dickson’s conjecture at present. Keywords Exponential Diophantine equation · Simple K 4 -group · Dickson’s conjecture · Zsigmondy’s theorem Mathematics Subject Classification 11D61 · 11D45 · 20D05
1 Introduction A finite simple group is called a K n -group if its order is divisible by exactly n different primes. By a classical pa q b theorem of Burnside, every group of order pa q b is solvable, where p and q are primes, and a and b are positive integers; hence, there is no simple K 2 -group. On the other hand, there are only eight different simple K 3 -groups [1]. In this paper, we concentrate on describing simple K 4 -groups.
Communicated by Mohammad Reza Darafsheh. This work was supported partially by the NSFC (No. 11671063) and Scientific Research Innovation Team Project Affiliated to Yangtze Normal University (No. 2016XJTD01).
B
Wujie Shi [email protected] Shaohua Zhang [email protected]
1
School of mathematics and statistics, Yangtze Normal University, Chongqing 408102, People’s Republic of China
2
Department of Mathematics, Chongqing University of Arts and Sciences, Chongqing 402160, People’s Republic of China
3
School of Mathematics, Suzhou University, Suzhou 215006, People’s Republic of China
123
Bulletin of the Iranian Mathematical Society
In the famous book Unsolved Problems in Group Theory, the following problem is asked: is the number of simple K 4 -groups finite or infinite? See [2]: Problem 13.65. This problem is the first to be posed by the second author Shi [3]. Denote by N and P the set of positive integers and the set of prime numbers, respectively. In [3], the second author claimed that the simple K 4 -group problem can be reduced to the four Diophantine problems: p 2 − 1 = 2a 3b q c , p, q ∈ P, p > 3, q > 3, a, b, c ∈ N, 2m − 1 = p, 2m + 1 = 3q n , p, q ∈ P, p > 3, q > 3, m, n ∈ N,
(1.1) (1.2)
3m − 1 = 2 p n , 3m + 1 = 4q, p, q ∈ P, p > 3, q > 3, m, n ∈ N, 3m − 1 = 2 p, 3m + 1 = 4q n , p, q ∈ P, p > 3, q > 3, m, n ∈ N.
(1.3) (1.4)
In 2001, Bugeaud et al. [4] showed that if n > 1, (1.2) and (1.4) have no solution and (1.3) has only the solution ( p, q, m, n) = (11, 61, 5, 2). In Sect. 2, we will prove that if c > 1, (1.1) has only the solutions ( p, q, a, b, c) = (97, 7, 6, 1, 2) and ( p, q, a, b, c) = (577, 17, 7, 2, 2). Our methods are slightly different from those in [4]. Thus, the infinitude of simple K 4 -groups can be decided by the following three Diophantine problems: p 2 − 1 = 2a 3b q, p, q ∈ P, p > 3, q > 3, a, b ∈ N,
(1.5)
2 − 1 = p, 2 + 1 = 3q, p, q ∈ P, p > 3, q > 3, m ∈ N, 3m − 1 = 2 p, 3m + 1 = 4q, p, q ∈ P, p > 3, q > 3, m ∈ N.
(1.6) (1.7)
m
m
By considering Diophantine equations (1.5), (
Data Loading...
