A Study About One Generation of Finite Simple Groups and Finite Groups

In this paper, we study the problem of how a finite group can be generated by some subgroups. In order to the finite simple groups, we show that any finite non-abelian simple group can be generated by two Sylow p1and p2-subgroups, where p1 and p2 are two different primes. We also show that for a given different prime numbers p and q, any finite group can be generated by a Sylow p-subgroup and a q-subgroup.


Introduction
Finite groups often arise when considering the symmetry of mathematical or physical objects, when those objects admit just a finite number of structure-preserving transformations. Important examples of finite groups include cyclic groups and permutation groups (Praeger & Schneider, 2018). During the twentieth century, mathematicians investigated some aspects of the theory of finite groups in great depth. One major area of study has been the classification of groups (Solomon, 2001). At the beginning of the 1980s, the development of the theory of finite groups culminated in the classification of the finite simple groups, an impressive and convincing demonstration of the strength of its methods and results (Kurzweil & Stellmacher, 2004).
The classification of finite simple groups, (Gorenstein, Richard & Solomon, 2018), i.e. groups with no nontrivial normal subgroup, was completed in 2004. As a consequence, the complete classification of finite simple groups was achieved, meaning that all those simple groups from which all finite groups can be built are now known (Aschbacher, 2004).
Via its importance, the concept of the generation of a finite simple group and a finite group has widespread attention along time, where generation of finite groups by suitable subsets has many applications to groups and their representations. For example, due to the motivation to study irreducible projective representations of the sporadic simple groups, in (Martino, Pellegrini & Zalesski, 2014) it was established a useful connection between the generation of groups by conjugate elements and the existence of elements representable by almost cyclic matrices. It is known that finite non-abelian simple groups are 2-generated (Steinberg, 1962). In (Aschbacher & Guralnick, 1984) authors showed that any sporadic simple group can be generated by an involution and another suitable element, recall that a sporadic group is one of the 26 exceptional groups found in the classification of finite simple groups (Wilson, 1998). In (King, 2017) it was shown that every finite non-abelian simple group is generated by an involution and an element of prime order. In (Burness, Liebeck, & Shalev, 2013) authors showed that any maximal subgroup of a non-abelian finite simple group is 4-generated or less and that this bound is best possible. In 2007, Slattery in his paper (Slattery, 2007) described computational methods to enumerate, construct and identify finite groups of square-free order. Also, in (Breuer, Guralnick, & Kantor, 2008) authors conjectured that any finite group is 3 2 −generated if and only if every proper quotient is cyclic. In (Burness, Guralnick, & Harper, 2021) it was reduced this conjecture to almost simple groups. Recently, in (Dietrich & Low, 2021) the authors, generalized Slattery's result to the class of finite groups that have cyclic Sylow subgroups and provide an implementation for the computer algebra system GAP (see https://www.gap-system.org/).
In this paper, in order to the finite simple groups, we show that any finite non-abelian simple group can be generated by two Sylow 1 -and 2 -subgroups, where 1 and 2 are two different primes. Also, we show that for a given different primes and , every finite group can be generated by a -subgroup and a -subgroup.
The paper is consists of an introduction and two fundamental sections. In Section 2, we study some generations of finite simple groups, we prove three lemmas, from which we conclude the fundamental theorem about the generation of finite simple groups by different Sylow subgroups. In Section 3, we set other fundamental results of the paper, concerning the generation of finite groups by some subgroups.

Generations of Simple Groups
For the fundamental result in this section (see Theorem 1, below), we need some auxiliary lemmas. Therefore firstly we will set three lemmas, with their proof. Lemma 1. Let = , ≥ 5. If ≤ is prime, then can be generated by a Sylow 2-subgroup and a Sylow -subgroup.
Proof. When ≤ 8 the result is clear. Suppose that > 8 and is an odd number. Let be a Sylow 2-subgroup of . The orbits of have distinct sizes and there are less than log 2 orbits. Let us choose a -element , such that has a cycle of length ≥ √ for odd and at least 2 ⁄ if = 2. In both cases, the size of the orbit is greater than the number of orbits of . Thus, we can choose a conjugate of so that an orbit of intersects every orbit of . ≔ 〈 , 〉 is transitive. Since is odd and ≤ , we find that is primitive. Since > 8, the only primitive subgroup of containing an element moving 4 points is (Wielandt, 1964), and therefore we reached the result. Now, suppose that > 9 is an even number. From the previous case, i.e. when is odd, we find that −1 = 〈 0 , 〉 for some -subgroup and Sylow 2-subgroup 0 . Let be a Sylow 2-subgroup of properly containing 0 . Then 〈 , 〉 properly contains 〈 0 , 〉 = −1 , whence 〈 , 〉 = . Thus we reached the proof of the Lemma 1. Lemma 2. Let be a sporadic simple group and let be a Sylow 2-subgroup of . If 1 ≠ ∈ , then = 〈 , 〉 for some ∈ . holds. The numbers [ ( ): ] can be read off from (Wilson, 1998). The Monster group is known to contain exactly five classes of maximal subgroups of odd index, of the structures 2 1+24 . 1 , 2 10+16 . 10 + (2), 2 2+11+22 . ( 24 × Lemma 3. Let be a finite simply connected quasisimple group of Lie type in characteristic . Let be a Sylow -subgroup of . If ∈ is a noncentral element of , then = 〈 , 〉 for some ∈ . Proof. By Tits's lemma (Seitz, 1973), we find that any maximal subgroup containing is a parabolic subgroup and there is precisely one parabolic subgroup containing in each conjugacy class of maximal parabolic subgroups. If has twisted rank 1, then the Borel subgroup is the unique maximal subgroup containing and so clearly no non-central conjugacy class is contained in the Borel subgroup, whence the result holds. So we may assume that the rank is at least 2. First, suppose that is a classical group. If = 2 ( ), then choose a basis for the natural module 1 , … , and assume that stabilizes the flag associated with this ordered basis. Since acts 2-transitively on 1-dimensional spaces, given any non-central element ∈ , we can choose ∈ so that −1 1 is not in the span of 1 , … , −1 and so is not in any parabolic subgroup containing . Now suppose is any other classical group with natural module . Since we are assuming that has twisted rank at least 2, dim ≥ 4. First, suppose that is not an even-dimensional spin group in dimension at least 8. Fix a Sylow -subgroup. Then the maximal parabolic subgroups correspond to totally singular spaces of distinct dimensions. Let 1 be a nonzero element of the 1-dimensional fixed space of . If n is the rank of , then there is an ordered linearly independent set 1 , … , so that any maximal parabolic subgroup containing is the stabilizer of the totally singular space = 〈 1 , … , 〉. Let be a non-central element of . Suppose that 1 = . If and 1 are linearly dependent, then just choose ∈ so that maps 1 to element not in . If is not in , then already is not in any of the parabolic subgroups containing . The last case is if 〈 1 , 〉 is a 2-dimensional totally singular subspace of . Then there exists ∈ fixing 1 and mapping to any singular vector in 1 ⊥ and since 1 ⊥ is not contained in (since dim ≥ 4), the result holds in this case. Next, suppose that is an even-dimensional spin group of rank . In the case, when we have an orthogonal module of dimension 2 , we choose a linearly independent set of basis vectors 1 , … , +1 so that the maximal parabolic subgroups containing are the stabilizer of the totally singular subspace = 〈 1 , … , 〉, = 1, … , − 2 of dimension and the two totally singular subspaces 〈 1 , … , −1 , 〉 and 〈 1 , … , −1 , +1 〉. Finally, let us suppose that is an exceptional group of twisted 6 rank . Let 1 , … , be the distinct maximal parabolic subgroups containing . Let = for a non-central element of . It follows that (Lawther, Liebeck & Seitz, 2002) Therefore ≠ ⋃ ( ⋂ =1 ), and therefore we finished the proof.
It remains to consider the special cases when is the simply connected group but ( ) ⁄ is not simple. We can ignore the case when is a solvable group. This leaves the groups 4 (2), 2 (2), 3 2 (3), and 2 4 (2).The first three cases have socle 6 , 3 (3) and 2 (8) and we have already proved the result for these groups. In the final case, the socle is the derived subgroup of index 2 but precisely the same previous argument (i.e. there are only two maximal subgroups containing a Sylow 2-subgroup which are parabolic subgroups intersected with the derived subgroup).
From previous lemmas, we reach the proof of the following fundamental theorem in this section.
Theorem 1. Let be a finite non-abelian simple group. Then there exists a prime 1 with divides | |, such that can be generated by a Sylow 1 -subgroup and a Sylow 2 -subgroup, where 2 any given prime divides | |.

Generations of Finite Groups
Recall that a subgroup of is called intravariant, if for any automorphism of , ( ) and are conjugate in . The main result in this section will be presented in the following theorem.
Theorem 2. Let and are different primes and a finite group. Then there exists a Sylow -subgroup of and an intravariant -subgroup such that = 〈 , 〉.
Proof: Let and be fixed primes, such that ≠ , and let be a counterexample of minimal order. Firstly we will prove that: ( ) = 1. Let us assume the converse and let 1 ≠ be a minimal characteristic subgroup of with an elementary abelian -group. By the minimality, we find that ⁄ = 〈 ⁄ , ⁄ 〉, where is a Sylow -subgroup of and is a subgroup of containing with ⁄ an intravariant -group of ⁄ . Since gcd(| | | | ⁄ , | |) = 1, it follows that 2 ( ⁄ , ) = 0, therefore = 1 with 1 a complement of in . In particular, 1 is a -subgroup. Suppose is an automorphism of . Then by conjugating by an element of , we may assume that ( ) = . Thus, ( 1 ) is a complement of in . Since 1 ( 1 , ) = 0, then all complements of in are conjugate via an element of and so ( 1 ) is conjugate to 1 , therefore 1 is intravariant and it clear that = 〈 , 1 〉.
At the second we will prove that: ( ) = 1. Let us assume the converse and let be a minimal characteristic subgroup of with a -subgroup . We have ⁄ = 〈 ⁄ , ⁄ 〉, where ⁄ is a Sylow -subgroup of ⁄ and ⁄ is intravariant -subgroup of ⁄ . Because is characteristic then is intravariant and clearly a -subgroup. It is clear that = , where is a Sylow -subgroup of and of . Thus = 〈 , 〉 and the result follows. Now, let be a minimal characteristic subgroup of . By the previous arguments, we find that = × … × , where is a non-abelian simple group of order divisible by . By Theorem 1, we find that = 〈 , 〉, where is a Sylow -subgroup of and is a Sylow -subgroup of for some ≠ . Let = ( ), then is a proper subgroup of , thus = 〈 , 〉, where is a Sylow -subgroup of and is an intravariant -subgroup of . Since normalizes , we assume that ≥ .
is the normalizer of a Sylow subgroup of a characteristic subgroup of and so and are intravariant. Assume that contains a Sylow -subgroup 1 of . Then 1 = for some ∈ . Then ≥ . We have the following fact = : = 〈 , 〉. Firstly we have ≥ 〈 , 〉 = . Since ∈ we see that = and so ≥ 〈 , , 〉 ≥ 〈 , 〉. By the Frattini argument, we find that = , thus = as we need.