A new characterization for some linear groups

A new characterization of A7 and A8

“…So it is natural to investigate the Thompson's Problem by |G| and nse(G). In [4], [2], [3] and [1], it is proved that all simple K 4 − groups, symmetric groups S r where r is prime, sporadic simple groups and L 2 (p) where p is prime, can be uniquely determined by nse(G) and the order of G. In [9] and [8], it is proved that the groups A 4 , A 5 and A 6 , L 2 (q) for q ∈ {7, 8, 11, 13} are uniquely determined by only nse(G). In [9], the authors gave the following problem:…”

A characterization of L3(4)

“…It follows that the group P 2 acts fixed point freely on the set of elements of order 7 and so |P 2 | | s 7 (= 8064). So we have |P 2 | | 2 7 . Therefore…”

NSE characterization of the Chevalley group $$\varvec{G}_{\varvec{2}} {\varvec{(4)}}$$ G 2 ( 4 )

“…Similarly, as |P 3 | = 9, we also get a contradiction. Now, we suppose that exp (P 3 ) = 9, according to Lemma 2.1 |P 3 ||1 + s 3 + s 9 . There are several cases for s 9 .…”

“…Now, we suppose that exp (P 3 ) = 9, according to Lemma 2.1 |P 3 ||1 + s 3 + s 9 . There are several cases for s 9 . Namely, 982800, 3354624, 22276800, 11980800, 15724800, 50319360, 15724800, 38707200, 67092480, and 23961600.…”