Australian Journal of Basic and Applied Sciences, 6(2): 130-132, 2012 ISSN 1991-8178
A New Characterization of The Janko Group
Seyed Sadegh Salehi Amiri, Alireza Khalili Asboei and Abolfazl Tehranian
Department of Mathematics, Science and Research Branch, Islamic Azad University, Tehran, Iran.
Abstract. Let be a finite group and be the set of elements of order . Let and be the number of elements of order in . Set nse( ):= | . It is proved that Mathieu groups are uniquely determined by its nse and order. In this paper as the main result, it is proved that if is a group such that nse( )=nse , then .
Key words: Janko group, element order, simple group, Sylow subgroup.
INTRODUCTION
If is an integer, then we denote by the set of all prime divisors of . Let be a finite group. Denote by the set of primes such that contains an element of order . Also the set of element orders of is denoted by . A finite group is called a simple group, if is a simple group with | | . Set =| | the order of is and nse : | . In fact, is the number of elements of order in and nse is the set of sizes of elements with the same order. In (Shao, C.G., et al., 2008) and (Shao, C.G., Q.H. Jiang. 2010), it is proved that all simple groups and Mathieu groups can be uniquely determined by nse( ) and order . In (Khatami, M. et al., 2009) and (Shen, R., et al., 2010), it is proved that if is one of the groups , , and 2, , for 7,8,11,13 , then it can be uniquely determined by only nse( ). In this paper, by new method we show that the Janko group is characterizable by only nse( ). In fact, the main theorem of our paper is as follow:
Main Theorem: Let be a group such that nse( ) = nse( ), then . We note that there are finite groups which are not characterizable by nse(G) and | |. In 1987, Thompson gave an example as follows: Let C C C C A and L 4 C be the maximal subgroups of . Then nse( ) = nse( ) and | |=| |, but . Throughout this paper, we denote by the Euler totient function. If is a finite group, then we denote by a Sylow subgroup of and : is the number of Sylow subgroup of , that is., =| |. All further unexplained notations are standard and refer to (Conway, J. H., et al., 1985), for example.
2. Preliminary Results: In this section we bring some preliminary lemmas to be used in the proof of main theorem theorem. Lemma 2.1. (Frobenius, G., 1895) Let be a finite group and be a positive integer dividing | |. If | 1 , then || |. Lemma 2.2. (Khosravi, B. and B. Khosravi. 2005) Let be a sporadic simple group and be the greatest element of . Then is uniquely determined by | | and n S). Lemma 2.3. (Shen, R., et al., 2010) Let be a group containing more than two elements. Let and be the number of elements of order in . If = | is finite, then G is finite and | | 1 . Let be a group such that nse(G) = nse( ). By Lemma 2.3, we can assume that is finite. Let be the number of elements of order . We note that = . , where is the number of cyclic subgroups of order in . Also we note that if 2, then φ is even. If , then by Lemma 2.1 and the above notations we have: