Order of product of elements of two subgroups

Let be a finite group. Let , , and where denotes a subgroup and . Let denote the (group or element) order of .

(a) Show that for every , where is the order of .

(b) Let such that is coprime to . Prove that if , then .

I'm not quite sure how to start on this problem. I don't want a full solution, just some help in figuring it out.

Re: Order of product of elements of two subgroups

Quote:

Originally Posted by

**thehobbit** Let

be a finite group. Let

,

, and

where

denotes a subgroup and

. Let

denote the (group or element) order of

.

(a) Show that for every

, where

is the order of

.

Here's my hint, all you have to do is show that , can you do this?

Quote:

(b) Let

such that

is coprime to

. Prove that if

, then

.

I'm not quite sure how to start on this problem. I don't want a full solution, just some help in figuring it out.

Let then and so for [tex]k\in K[/tx] and and so and so you know that ...

Re: Order of product of elements of two subgroups

I thought about this one while shopping and now the answer seems quite simple.

For (a), since and , we have , thus

For (b), picking up where you left off, . Since is coprime to , we must have , hence is the trivial subgroup, yielding the desired result.

Re: Order of product of elements of two subgroups

Quote:

Originally Posted by

**thehobbit** I thought about this one while shopping and now the answer seems quite simple.

For (a), since

and

, we have

, thus

For (b), picking up where you left off,

. Since

is coprime to

, we must have

, hence

is the trivial subgroup, yielding the desired result.

Right!

Re: Order of product of elements of two subgroups