Results 1 to 5 of 5

Thread: Minor Confusion Over Subgroup Index

  1. #1
    Newbie
    Joined
    Sep 2009
    Posts
    16

    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
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor Drexel28's Avatar
    Joined
    Nov 2009
    From
    Berkeley, California
    Posts
    4,563
    Thanks
    22
    Quote Originally Posted by Keilan View Post
    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 $\displaystyle 2\mathbb{Z}\cup \left(2\mathbb{Z}+1\right)=\mathbb{Z}$
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    Sep 2009
    Posts
    16
    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.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor Drexel28's Avatar
    Joined
    Nov 2009
    From
    Berkeley, California
    Posts
    4,563
    Thanks
    22
    Quote Originally Posted by Keilan View Post
    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.
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Newbie
    Joined
    Sep 2009
    Posts
    16
    Ah, I see. Seems like I am constantly getting tripped up by notation in this whole group and ring theory business.

    Thanks for your help!
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. proof that subgroup has finite index
    Posted in the Advanced Algebra Forum
    Replies: 6
    Last Post: Mar 10th 2010, 10:33 PM
  2. Subgroup index
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: Nov 22nd 2009, 09:20 PM
  3. Subgroup of index 2 is normal
    Posted in the Advanced Algebra Forum
    Replies: 5
    Last Post: Nov 7th 2009, 03:08 PM
  4. Subgroup with prime index
    Posted in the Advanced Algebra Forum
    Replies: 3
    Last Post: Mar 29th 2009, 03:19 PM
  5. Group, subgroup H of index n that is not normal
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: Nov 25th 2008, 04:29 AM

Search Tags


/mathhelpforum @mathhelpforum