On Simple K4-Groups

Pure quantitative characterization of linear groups over the binary field

“…(1)-(3), and it is even not easy to determine whether they have finite or infinite integer solutions. Recently, this problem was partially solved by Bugeaud et al [14] who proved that if the exponent b > 1 or c > 1, then the Diophantine equations (2) and (3) have no solution except for the case of L 2 (3 5 ). However, if the exponent b or c is equal to one, it is more difficult to solve eqs.…”

“…Since then, many simple groups have been characterized by using similar approaches, e.g., L 2 (q) with q = 9, L 3 (2 m ) with m 2, U 3 (2 m ) with m 2, the Suzuki-Ree simple groups, all sporadic simple groups except J 2 and some non-soluble groups [2]. In [3], Darafsheh et al proved that L 5 (3), A 14 and their automorphism groups can be characterized by the sets of the element orders. The same method was used by Kondratév and Mazurov [4] to characterize the alternative groups of the prime degree.…”

On 9- and 10-decomposable finite groups

“…We first assume that ψ(G ) = 1. Then by [26], G is isomorphic to PSL(2, q), q = 5, 7, 8,9,11,13,16,PSL(3,4), Sz (8), PSL(2, 3 n ), where 3 n −1 2 and 3 n +1 4 are primes, or PSL(2, 2 n ), where 2 n − 1 and 2 n +1 3 are primes. By Table 1, we must only consider the cases that G ∼ = PSL(2, 3 n ), PSL(2, 2 n ).…”

$(2,m,n)$-groups with Euler characteristic equal to $-2^as^b$

“…In particular, if χ = −2 a s b t c then the group G may have a chief factor isomorphic to T k for some simple group T and k > 1; in this case |T | must be divisible by exactly three primes (namely 2, s and t) and such groups do exist. (There are precisely eight simple groups whose orders are divisible by exactly three primes, namely A 5 , A 6 , PSp 4 (3), P SL 2 (7), P SL 2 (8), P SU 3 (3), P SL 3 (3) and P SL 2 (17); this fact is not dependent on the classification of finite simple groups; see, for example, [BCM01]. )…”