Applications of Sylow's Theorem

[msd213]

Homework Statement

Show the G is not simple whenever one of the following is true

(i) $$|G|=2^mp^n$$ where $$2^k \neq 1$$ (mod p) for any $$1 \leq i \leq m$$.
(ii) $$|G|=p^nq$$, where $$p \neq q$$ (primes) and $$q \neq 1$$ (mod p)
(ii)$$|G|=ap$$, where $$p \not | a$$ and $$(kp+1) \not | a$$ for any $$k \in \mathbf{N}$$

Homework Equations

The Attempt at a Solution

I know that these all involve the use of Sylow's theorem but I'm stuck; I think I might just need the smallest of hints to move forward.

 

[kurt.physics]

(i) Since G has order 2^mp^n, by Sylow's theorem there must be p^n-1 subgroups of order p. This means that for G to be simple, the number of Sylow p-subgroups needs to equal 1. Since 2^k \neq 1 (mod p), the number of Sylow p-subgroups cannot equal 1, so G cannot be simple. (ii) Since G has order p^nq, by Sylow's theorem there must be q-1 subgroups of order p. This means that for G to be simple, the number of Sylow p-subgroups needs to equal 1. Since q \neq 1 (mod p), the number of Sylow p-subgroups cannot equal 1, so G cannot be simple. (iii) Since G has order ap, by Sylow's theorem there must be a-1 subgroups of order p. This means that for G to be simple, the number of Sylow p-subgroups needs to equal 1. Since (kp+1) \not | a for any k \in \mathbf{N}, the number of Sylow p-subgroups cannot equal 1, so G cannot be simple.