D_p(3) (p ≥5) can be characterized by its order components

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

“…which by Lemma 9 in [23] we deduce r = 2. Now it is easy to see that for q = 2 f the equation q n = 2 p+1 − 3 is impossible.…”

“…, where p is a prime number and 5 < p = 2 m − 1, in [3], [6] and [23], respectively. Some projective special linear (unitary) groups have been characterized in [11], [13], [14], [15] and [17].…”

“…It can be verified that if p ≥ 5, then (3,4). From now on we use Lemma 9 in [23]. According to this lemma if r is an odd prime divisor of a = n i=1 (2 2i − 1), then the r-part of a, i.e., a r , satisfies a r < 2 3n and moreover if r ≥ 5, then a r < 2 2n .…”

“…Using the following inequalities together with Lemma 9 in [23] and Lemma 6 the second case is ruled out.…”

“…4(p+1) which enables us to use Lemma 6 of [23] and Lemma 3 to deduce r = 2 or 3 in the respective cases. Then using the method of Case 11 one is able to rule out this case.…”