Understanding the < > Notation for Subgroups: A Closer Look

[Gear300]

I'm having a bit of a tough time interpreting <S> for a set S. I know for an element a, <a> is the set of all integral powers of a with respect to a given operation, but for a set S = {a, b, c}, what would <a, b, c> turn out as?

Edit: The source of my trouble is with this: The subgroup <9, 12> of the group of integers with addition as the operation contains 12 + (-9) = 3 (in order for it to be a group). Here is what the text says: "Therefore <9, 12> must contain all multiples of 3." I thought <9, 12> would only consist of multiples of 9 and 12, but apparently, there is more to it.

 

[Elucidus]

Not knowing more about the group in question, I am assuming that the group operation is addition. Then $\langle 9, 12 \rangle$ is going to be all linear combinations of 9 and 12, i,.e. $9m+12n$ for integers m and n. It turns out that that this is identical to the set of all multiples of 3.

I suspect that for any two intgers a and b that $\langle a,b \rangle = \langle \gcd (a,b) \rangle$.

--Elucidus

 

[Gear300]

Thanks for the reply...it seems as though it never came to me anywhere in the text. I was thinking the notation for <a₁, a₂ ... a_n> was simply the set of all integral powers of the elements.

So from this, I'm assuming that < > with respect to addition can be interpreted as a linear combination between elements in the group, right?

 

[g_edgar]

Are you sure < > notation was never defined in that book? From your comments, I am guessing that (for that book) <S> means the subgroup generated by S.

 

[Gear300]

Are you sure < > notation was never defined in that book? From your comments, I am guessing that (for that book) <S> means the subgroup generated by S.

Yup...that was part of its definition. The definition it gave was: <a> is the set of all integral powers of a for a given operation. They then went into further analysis. I just wasn't sure what <S> of a set S = {a, b, c, ...} was since the definition they gave was for a single element a.