On finite factorizable groups

A Characterization of the Finite Simple Groups

“…(1), (3), and (2)(c) are well known (see [13, I, p. 190 and II, p. 502] and [1,14,32]). Let n = 2 r k, where 2 k = 1.…”

“…It is obvious that M p G and m p G are, respectively, the largest and the smallest indices of the maximal parabolic subgroups of G. Essentially, we can determine all the maximal parabolic subgroups of G from the Dynkin diagram of G 3 6 and calculate precisely their indices from the information in [1,3,7,8,15], and obtain the proof of (2) First we prove (1) in Lemma 4. Suppose that (1) is not true, that is, suppose there exists a maximal subgroup K of G such that G K = s < M p G , but s is neither the index of a maximal parabolic subgroup of G nor in Table II and Pot p s < t n (see Table II for t n ).…”

“…By Lemma 4, π t N ⊂ π t G . If m > 2, B m q has the maximal parabolic subgroup P 2 such that 1 . Suppose P N = P G .…”

“…According to the classification theorem of the finite simple groups, from now on we let N be a Lie-type simple group of characteristic p 1 over GF q 1 , where q 1 = p f 1 1 . Now we prove that Theorem 1 holds if p = p 1 .…”

Теоремы Силовского Типа

“…3 Таким образом, с проблемой 1.8 тесно связан сле-дующий вопрос. Изучением этой проблемы занимались многие математики (см., например, [9], [15], [34]- [36], [38]- [45], [49], [55], [70]- [73]). Важность проблемы 3.2 бы-ла осознана уже Ф. Холлом.…”