Recognition of alternating groups of prime degree from their element orders

New characterization of symmetric groups of prime degree

“…Williams and Kondrat'ev (see [11,4,5]), proved a series of important results on prime graphs. We apply these results to prove Theorem A.…”

“…According to the classification theorem of finite simple groups and [4,5,11,12], we can list the order components of finite simple groups with disconnected prime graphs as in Tables 1-3 in [1].…”

“…Now we state the following lemma which is proved in [5,Lemma 6], with some differences and classify the simple groups of Lie type with prime odd order component by θ function, which is introduced later.…”

CHARACTERIZATION OF THE GROUPS Dp+1(2) AND Dp+1(3) USING ORDER COMPONENTS

“…By Lemma 1 in [22] there are at least three distinct primes p i such that 2 p−1 < p i < 2 p − 1. Of course we have p i | |A p |.…”

On the Prime Graph of Simple Groups

“…In this paper, we will use this result to solve the above problem; our main result is Corollary 4 below. This follows from our first theorem, which treats the case where H is a maximal subgroup of G. (In the final column of Table 1, we record the number of connected components in Γ(G), denoted by s(G), which is taken from [11,.) Theorem 1.…”