STUDY ON GROUP OF ORDER p-(n) q-(m)
q이고 P와 q가 서로소일 때, 위수가 P^n·q인 군은 가해이다. (g) P와 q가 서로소일 때, 위수가 P^(2)·q인 군은 가해이다. (h) P와 q가 서로소일 때, 위수가 P^(2)·q^(2)인 군은 정규 Sylow 부분군을 갖는다. (i) P와 q가 서로소일 때, 위수가 P^(2)q^(2)인 군은 가해이다.;In this paper, we will study the properties and applications of finite groups of order p^(n). q^(m), where p and q are distinct primes and m and n are natural numbers. We will show the following main facts in this paper; (a) Every finite p-group has a nontrivial center of at least p elements. (b) Every finite p-group is solvable. (c) For a prime number p, every group G of order p^(2) is abelian. (d) If p and q are distinct primes with p< q, then every group G of order p.q has a single subgroup of order q and this subgroup is normal in G. Hence G is not simple. If q is not congruent to 1 modulo p, then G is abelian and cyclic. (e) The group G of order pq, where p and q are distinct primes, is solvable. (f) There is no simple group of order p^(r)·m, where p is a prime and m < p. (g) The group G of order p^(n)·q, where p and q are distinct prlmes and p >q, is solvable. (h) The group G of order p^(2)·q, where p and q are distinct primes, is solvable. (i) If G is a group of order p^(2)q^(2) with p and q distinct primes, then G has a normal Sylow subgroup. (j) Every group G of order p^(2)q^(2) is solvable, where p and q are distinct primes.