Need help with a proof - subgroups

Hi guys.

I need to prove the following statment, I did most of the work (which I'm not entirely sure is valid - so a feedback would be great), but I'm having some trouble proving the last part of it, and could really use some tips here.

Let be a group.

Let be non-trivial subgroups of ( ).

Prove that .

Proof:

If we're done, since , and , we get that there exist s.t .

If , without loss of generality, suppose that .

Let s.t (there exist such since isn't trivial and ).

Let .

Obviusly , so If we show that , we're done.

Let's assume by contradiction that .

It then follows that .

Case ( ):

If , then , for some .

But then:

but ! that contradics our assumption of .

Case ( ):

I have nothing here...

any help would be greatly appreciated!

Re: Need help with a proof - subgroups

Quote:

Originally Posted by

**Stormey** Hi guys.

I need to prove the following statment, I did most of the work (which I'm not entirely sure is valid - so a feedback would be great), but I'm having some trouble proving the last part of it, and could really use some tips here.

Let

be a group.

Let

be non-trivial subgroups of

(

).

Prove that

.

Proof:

If

we're done, since

, and

, we get that there exist

s.t

.

If

, without loss of generality, suppose that

.

I see reason that K would have to be a subgroup of H. Consider the symmetries of a square, D_8. H = {e, reflection over the x - axis} and K = {e, reflection over the y - axis} are both subgroups of D_8, but K is not a subgroup of H.

-Dan

Re: Need help with a proof - subgroups

Quote:

Originally Posted by

**topsquark** I see reason that K would have to be a subgroup of H. Consider the symmetries of a square, D_8. H = {e, reflection over the x - axis} and K = {e, reflection over the y - axis} are both subgroups of D_8, but K is not a subgroup of H.

-Dan

Hi, topsquark, thanks for the help.

Yes of course, K does not have to be a subgroup of H.

but there has to be an element in H that doesn't appear in K (or vice versa).

so besides this distinction, the rest of the proof is valid.

(correct me if I'm wrong)

Re: Need help with a proof - subgroups

[QUOTE=Stormey;804944]

Let be a group.

Let be non-trivial subgroups of ( ).

//

Re: Need help with a proof - subgroups

Quote:

Originally Posted by

**Stormey** Let

be a group.

Let

be non-trivial subgroups of

(

).

Prove that

.

I am confused by the notation.

I would assume that ( means that each of are **proper** subgroups.

If that is the case, then it must be true that neither is a subset of the other.

So

Now can you argue that BUT .

Did I miss-understand that notation?

Re: Need help with a proof - subgroups

Quote:

Originally Posted by

**Plato** I am confused by the notation.

I would assume that (

means that each of

are

**proper** subgroups.

If that is the case, then it must be true that neither is a subset of the other.

So

Now can you argue that

BUT

.

Did I miss-understand that notation?

No, you got it. that is indeed the meaning of this notation.

Why can I argue that ? how do I justify it?

Re: Need help with a proof - subgroups

Quote:

Originally Posted by

**Stormey** No, you got it. that is indeed the meaning of this notation.

Why can I argue that

? how do I justify it?

That is the very meaning of , surely you know that.

If so neither is the subset of the other.

If .

Same for

Re: Need help with a proof - subgroups

ohh...

So I actually have three cases:

I: where .

II: where .

III where .

Thank you, again. (Happy)