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Group theory: cyclic groups question?

  • 17-03-2013 2:04pm
    #1
    skyscraperblue
    Registered Users, Registered Users 2 Posts: 747 ✭✭✭


    Hi there,
    I'm having huge trouble with my group theory homework assignment. I don't know why but I find it extremely difficult to apply the theory of cyclic groups to any actual group - I know in theory that it's generated by one element but when I try to find that element I get confused. My questions ask me to determine whether the following groups are cyclic:
    • Z3 under addition
    • Z6* under multiplication
    • the direct product Z2 x Z2
    Then I have to find all the subgroups of these groups. Please help!


Comments

  • #2
    TheBody
    Registered Users, Registered Users 2 Posts: 5,633 ✭✭✭TheBody


    skyscraperblue wrote: »
    Hi there,
    I'm having huge trouble with my group theory homework assignment. I don't know why but I find it extremely difficult to apply the theory of cyclic groups to any actual group - I know in theory that it's generated by one element but when I try to find that element I get confused. My questions ask me to determine whether the following groups are cyclic:
    • Z3 under addition
    • Z6* under multiplication
    • the direct product Z2 x Z2
    Then I have to find all the subgroups of these groups. Please help!

    Why don't you start with the first one: [latex]\mathbb{Z}_3[/latex]. The elements of this group are {0, 1, 2}. So under addition in [latex]\mathbb{Z}_3[/latex], will any element in this group generate all these elements if added to itself a number of times? For example, clearly 0 won't be a generator for the group because no matter how many times we add 0 to itself, we will never generate the elements in the group. Try the different elements and let us know what you get.


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