Answer to Question #287598 in Abstract Algebra for Amay

Question

Let g be a finite non abelian group of order n with the property that G has a subgroup of order K for each positive integer K dividing n. prove that G is not a simple group

Answer 1

If n is prime then G is abelian, so n can't be prime.

If n is not prime, let $n=p_1^{e_1}\times ...\times p_m^{e_m}, m\geq2$

(prime decomposition of n)

Now,

$p_1|n\Rightarrow \text{G has a subgroup, S of order $p_1$}\\ \text{As $p_1$ is prime, so S is abelian.} \Rightarrow S\neq G\\ p_1\neq1\Rightarrow \text{S is a proper non-trivial subgroup of G}\\~\\ \therefore \texttt{G is not simple }~~~~~~~~~~~~~~~~~~~~\text{(Proved)}$