# Thread: Minor Confusion Over Subgroup Index

1. ## Minor Confusion Over Subgroup Index

So, I'm reading in Algebra by Serge Lang and I have come to the definition that the index of H in G, (G:H) is the number of left cosets of H in G. I'm having trouble figuring this out.

Take for example,
G = Z (the set of integers)
H = 2Z

Now, I know that (G:H) = 2 from various examples, but I don't understand why. What are the two cosets of 2Z that form H?

By my definition, a coset of H is a subset of G of form aH where a is from G. But for any a in Z, a(2Z) = 2Z. So how do we get two cosets?

Thanks

2. Originally Posted by Keilan
So, I'm reading in Algebra by Serge Lang and I have come to the definition that the index of H in G, (G:H) is the number of left cosets of H in G. I'm having trouble figuring this out.

Take for example,
G = Z (the set of integers)
H = 2Z

Now, I know that (G:H) = 2 from various examples, but I don't understand why. What are the two cosets of 2Z that form H?

By my definition, a coset of H is a subset of G of form aH where a is from G. But for any a in Z, a(2Z) = 2Z. So how do we get two cosets?

Thanks
You always know given $\displaystyle H\leqslant G$ that $\displaystyle H$ itself will be one of the subgroups. Now, $\displaystyle 2\mathbb{Z}+1=\left\{\cdots,-3,-1,1,3,\cdots\right\}$ and you're done. Why? Because $\displaystyle 2\mathbb{Z}\cup \left(2\mathbb{Z}+1\right)=\mathbb{Z}$

3. But is 2Z + 1 a coset of H?

That is the confusing part to me. I was under the impression that a coset of H had the form aH, not a + H.

4. Originally Posted by Keilan
But is 2Z + 1 a coset of H?

That is the confusing part to me. I was under the impression that a coset of H had the form aH, not a + H.
Aha! Never fear...almost every person starting group theory has this trouble (myself included).

Really, that's bad notation but a necessary evil.

You'll remember that a group $\displaystyle (G,*)$ is a non-empty set $\displaystyle G$ along with an associative binary operation (function) $\displaystyle *:G\times G\to G$ such that there is some $\displaystyle e\in G$ with $\displaystyle *(e,g)=*(g,e)=g$ for all $\displaystyle g\in G$. For each $\displaystyle g\in G$ there is some $\displaystyle h\in G$ such that $\displaystyle *(g,h)=*(h,g)=e$. And closure (but that is implicit in how we defined $\displaystyle *$.

That said NO ONE wants to keep writing $\displaystyle *$ everytime. So, instead of $\displaystyle *(g,h)$ we write $\displaystyle gh$. This does not mean multpilcation. This is just short hand notation for the function $\displaystyle *$. Remember, $\displaystyle *$ can be many, many things (function composition, multplication, matrix multiplication, addition, etc.). Now, you'll also remember that in general $\displaystyle *(g,h)=gh\ne hg=*(h,g)$ and groups for which all the elements satisfy that aer called abelian. Now, it is customary (don't ask me why) to switch from using the short hand notation $\displaystyle gh$ to $\displaystyle g+h$ even though it means the same thing when the group is abelian.

So, it is true that a coset is of the form $\displaystyle gH$ but that really means $\displaystyle \left\{*(g,h):h\in H\right\}$ and for the group $\displaystyle \mathbb{Z}$ we have that $\displaystyle *$ means addition.

5. Ah, I see. Seems like I am constantly getting tripped up by notation in this whole group and ring theory business.