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Math Help - Minor Confusion Over Subgroup Index

  1. #1
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    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
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  2. #2
    MHF Contributor Drexel28's Avatar
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    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 H\leqslant G that H itself will be one of the subgroups. Now, 2\mathbb{Z}+1=\left\{\cdots,-3,-1,1,3,\cdots\right\} and you're done. Why? Because 2\mathbb{Z}\cup \left(2\mathbb{Z}+1\right)=\mathbb{Z}
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  3. #3
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    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.
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  4. #4
    MHF Contributor Drexel28's Avatar
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    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 (G,*) is a non-empty set G along with an associative binary operation (function) *:G\times G\to G such that there is some e\in G with *(e,g)=*(g,e)=g for all g\in G. For each g\in G there is some h\in G such that *(g,h)=*(h,g)=e. And closure (but that is implicit in how we defined *.

    That said NO ONE wants to keep writing * everytime. So, instead of *(g,h) we write gh. This does not mean multpilcation. This is just short hand notation for the function *. Remember, * can be many, many things (function composition, multplication, matrix multiplication, addition, etc.). Now, you'll also remember that in general *(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 gh to 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 gH but that really means \left\{*(g,h):h\in H\right\} and for the group \mathbb{Z} we have that * means addition.
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  5. #5
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    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!
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