in this paper, we generalize some transfer theorems.~in particular, we derive one of the main results of gagola(contemp math 524:49--60, 2010) from our results.

In this paper, without using the classification of finite simple groups, we determine the structure of  finite simple groups whose Sylow 3-subgroups are of the order 9. More precisely, we classify finite simple groups whose Sylow 3-subgroups are elementary abelian of order 9.

First we will prove a small lemma. Lemma 1. Let P be a p-Sylow subgroup of G. Let N be a normal subgroup of G such that P ⊂ N . Then all p-Sylow subgroups are in N . Proof. Let P̃ be a p-Sylow subgroup that is not P . Then by the Sylow Theorems, we know that gPg−1 = P̃ for some g ∈ G. Since P ⊂ N , and N is normal, we know that gPg−1 ⊂ N and hence P̃ ⊂ N . Thus all p-Sylow subgroups are in N . Now...

Solution. The prime factorization of 185 is 5*37. Given a group of order 185, Let n5 be the number of subgroups of order 5 and n37 the number of subgroups of order 37. Since subgroups of order 5 and 37 will be Sylow subgroups, by the Sylow Theorems n5 and n37 have to be 1 (mod 5) and 1 (mod 37), respectively and divide 37 and 5, respectively. Hence, n5 = n37 = 1 and the 5 and 37 Sylow subgroups...

Proof. Any element of odd order in a symmetric group is an even permutation, so the 3-Sylow and 5-Sylow subgroups of S5 lie in A5. Therefore it suffices to focus on A5. Since |A5| = 60 = 22 · 3 · 5, the 3-Sylow subgroups have size 3 and the 5-Sylows have size 5. Call the numbers n3 and n5. By Sylow III, n3 | 20 and n3 ≡ 1 mod 3, so n3 = 1, 4, or 10. The number of 3-cycles (abc) in A5 is 20, and...

In the Sylow theorems f we learn that if the order of a group 2Í is divisible hj pa (p a prime integer) and not by jo*+1, then 31 contains one and only one set of conjugate subgroups of order pa, and any subgroup of 21 whose order is a power of p is a subgroup of some member of this set of conjugate subgroups of 2Í. These conjugate subgroups may be called the Sylow subgroups of 21. It will be o...

Sylow subgroups are fundamental in the design of asymptotically efficient group-theoretic algorithms, just as they have been in the study of the structure of Ž . finite groups. We present efficient parallel NC algorithms for finding and conjugating Sylow subgroups of permutation groups, as well as for related problems. Polynomial-time solutions to these problems were obtained more than a dozen ...