# Minor Confusion Over Subgroup Index

#### 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

#### Drexel28

MHF Hall of Honor
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 $$\displaystyle 2\mathbb{Z}\cup \left(2\mathbb{Z}+1\right)=\mathbb{Z}$$

• Keilan

#### 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.

#### Drexel28

MHF Hall of Honor
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.

• Keilan

#### Keilan

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