A new recognition of some finite simple groups

Author
Ahmad -
Department of mathematics, Payame noor university, Po.Box:19395-3697
Abstract
Introduction

In this paper G is considered to be a finite group. We denote the set of elements order and the set of prime divisors of order G by πeG and π(G) , respectively. The largest element order of G is denoted by k1 (G), and the prime graph of G is denoted by Γ(G) , where two vertices u and v are adjacent if uv ∈πeG . After classification of finite simple groups, the problem that came to the researchers’ attention was the problem of recognizing a group with a specific characteristic. Properties such as elements order, the set of elements with the same order, graphs, etc. In fact we say the group G by property M is recognizable, whenever by isomorphic G be the only group by property M. The other methods are group recognition by using the order of the group and the largest element order. In other words, we say the group G is recognizably by using the order of the G and the largest element order whenever there exists group H so that ∣G∣= ∣H∣ and k1G=k1 (H) then G≅ H. It is known that some of groups are recognizable by this method. In this paper, we prove that the Stienberg group 3D4(2n ), where 24n-22n+1 is a prime number are recognizable by using the order of the group and the largest element order. In other words, we have the following main theorem.

Main Theorem

Let G be a group with the Steinberg group 3D4(2n ), where24n-22n+1 is a prime number such that ∣G∣= ∣ 3D4(2n )∣ and k1G=k1 ( 3D4(2n )), then G≅ 3D4(2n ).

Material and methods

In this research we prove that Steinberg group 3D4(2n ), where 24n-22n+1 is a prime number by using the order of the group and the largest element order. In order to prove the main theorem, we used Lemmas 4.2, 7.2 of the reference [18].



Results and discussion

In this section we prove the main result of this article. For simplicity the Steinberg simple group and prime number are denoted by D and p respectively. As mentioned in the previous section to prove the main result of this article we use the Lemma 4.2 of [18]. We prove p is an isolated vertex of prime graph. Using Lemma 4.2 we prove that G neither a Frobenius nor 2-Frobenius group. And for the case c this Lemma is satisfied. In other words, G has a normal series such that H and G/K and K/H are non-abelian simple groups. Moreover, H is a nilpotent group. Every odd components of prime is an odd component of the prime graph. In the next step, by using Lemma 7.2, since (5, ∣G∣)=1, we consider the groups of this Lemma. We also prove that isomorphism K/H≇L2 (q) , L3 (q), U3(q) , G2q, 2G2(q ), are a contradiction. Finally we have K/H ≅ 3D4(2n ). The proof be completed.

Conclusion

We conclude that in addition to a previously known criterion(test) for Steinberg groups recognition by their 2-sylow subgroups(ANTHONY HUGHES, CHARACTERIZATION OF 3D4 (q3), q = 2n BY ITS SYLOW 2-SUBGROUP, Proceedings of the Conference on Finite Groups,1976, Pages 103-105) and also the same order components (Guiyun Chen, Characterization of 3D4(q), Southeast Asian Bulletin of Mathematics, 25, pages 389–401,2002). Next, in this paper we can recognize them by the order of the group and the largest element order of the group.
Keywords
element order
the largest elements order
Stienberg group
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Mathematical Research
Volume 9, Issue 1 - Serial Number 24
Spring 2023
Pages 108-118