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Find possible subgroups given some elements

  1. Dec 6, 2016 #1
    1. The problem statement, all variables and given/known data

    Suppose that H is a subgroup of Z under addition and that H contains 2^50 and 3^50. What are the possibilities for H?

    2. Relevant equations

    Relevant concepts are just the definitions for a group and subgroup.

    https://en.wikipedia.org/wiki/Group_(mathematics)

    https://en.wikipedia.org/wiki/Subgroup

    3. The attempt at a solution

    The solution I was given is the following:

    V4SNMuS.png

    But what I'm wondering about, and would appreciate an answer to, is the following:

    But the subgroup has operation "addition" so 2^50 = 50*2 = (2*5*5)*2, and 3^50 = 50*3 = (2*5*5)*3, so the possibilities for H are:

    H=Z, H=2Z, H=10Z, H=25Z, H=50Z.

    Or am I missing something?
     
  2. jcsd
  3. Dec 6, 2016 #2

    fresh_42

    Staff: Mentor

    Firstly, after ##"H## contains ##\mathbb{Z}"## you can stop, because ##\mathbb{Z} \subseteq H \subseteq \mathbb{Z}## (the latter for being a subgroup) already implies ##H = \mathbb{Z}##.

    What you are missing is, that ##2^{50} \neq 50^2## or ##100##.
     
  4. Dec 6, 2016 #3
    Thanks for reply. But if we write out the notation, then, for group with operation "addition":

    250 = 50*2, and 502=2*50, and Z is commutative? That is, in general, an=n*a
     
  5. Dec 6, 2016 #4

    fresh_42

    Staff: Mentor

    No, you can't confuse these notations. ##2^{50}## is simply a number and the subgroup is ##(H,+)=2^{50}\cdot (\mathbb{Z},+) + 3^{50}\cdot (\mathbb{Z},+)##. Otherwise one would have defined ##H=100\mathbb{Z}+150\mathbb{Z}=50\mathbb{Z}##.
     
  6. Dec 7, 2016 #5

    Stephen Tashi

    User Avatar
    Science Advisor

    I think the notation used in the problem isn't consistent with the interpretation of ##a^k## for a group element ##a##. You are correct that for a group operation denoted "##*##", the interpretation of the notation ##a^3## is usually ##a*a*a##. So for the operation "##+##", the interpretation would be "##a+a+a##". However, I think the problem is using the notation for exponents in the standard sense that it is used for the field of real numbers. By that notation ## 2^3 = (2)(2)(2)## instead of ##2+2+2##.
     
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