Proving Subgroup Inclusions in Group Theory | Homework Help

[ashina14]

Homework Statement

Here's the question:
http://assets.openstudy.com/updates...3c487dfb-ceb105-1360099849081-grouptheory.png

Homework Equations

The Attempt at a Solution

Step 1: <S,T> subset <<S>,T> subset <<S>,<T>> (easy)
and Step 2: <S,T> subset <S,<T>> subset <<S>,<T>> (easy)
and Step 3: <<S>,<T>> subset <S,T> (slightly trickier)

I've done step 1 and 2 by saying <S> is smallest subgroup of G containing all elements of S and similarly with T.

No idea how to do step 3 i.e. how to show <<S>,<T>> subset <S,T> ??

 

[mighty2000]

Thank you for your question. Step 3 can be proven by using the definition of a subgroup.

First, we know that <S> is the smallest subgroup of G containing all elements of S. This means that <S> is a subgroup of G and contains all elements of S.

Similarly, <T> is also a subgroup of G containing all elements of T.

Now, let's consider the subgroup <S,T>. By definition, <S,T> is the smallest subgroup of G containing all elements of S and T. This means that <S,T> is a subgroup of G and contains all elements of S and T.

Therefore, we can conclude that <S,T> is a subgroup of both <S> and <T>, which means that <S,T> is a subgroup of the subgroup <S,T>. Hence, we have shown that <<S>,<T>> subset <S,T>.

I hope this helps to clarify the solution for step 3. If you have any further questions, please don't hesitate to ask.
Scientist