Exploring Subgroups of Finite Nonabelian Groups: Order and Cyclic Structure

[sza772250]

let G be a finite nonabelian group and h be a subgroup of G. suppose that the order of H is p where p is a prime
(1)prove that h is abelian
(2)How many elements of h have order P?
(3)Suppose K is a subgroup of G with order q, where q is a prime and q doesn't equal p. what is H intersect K?

for(1) to prove h is abelian, I can show H is cyclic, but how to show H is cyclic?(because order of p is prime?)
I have no idea how to do (2) and (3)

 

[spamiam]

This looks like an exercise in Lagrange's Theorem: http://en.wikipedia.org/wiki/Lagrange's_theorem_(group_theory).

Your strategy for (1) looks good, and this theorem will help you prove it.

 

[Deveno]

(1) consider the order of a non-identity element h of H. what orders are possible for h?

what does that tell us about the order of <h>?

(2) how many non-identity elements does H have? do they all have the same order?

(3) suppose x is in H, and x is in K. what can we say about the order of x? are there any divisibility arguments you can use?