Group theory: cyclic groups question?

skyscraperblue

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!

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.