Re: Self Study problem help - Group theory

On Tue, 26 Jul 2005 15:51:09 -0500, mstemper@xxxxxxxxxxxxxxxx (Michael
Stemper) wrote:

>In article <dc66hq$sop$0@xxxxxxxxxxxx>, Ignoramus5833 writes:
>
>>what about a group that is made of 1/2, 1/3, and all sums and
>>differences thereof. For example, 0=1/2-1/2, 5/6 = 1/2+1/3, and so on
>>and so forth.


>(1/2) - (1/3) = (1/6)
>
>So, this group would contain:
> ... (-4/6), (-3/6), (-2/6), (-1/6), (0/6), (1/6), (2/6), (3/6) ...
>
>I don't think that it has any other elements.
>
>This looks suspiciously like the additive group of integers, going incognito.

Right, it's an an infinte cyclic group generated by 1/6 (or -1/6, take
your pick) and yes, it is, in essence, the additive group of integers.
A more precise statement is that the subgroup <1/6> (the subgroup
generated by 1/6) is isomorphic to the additive group of integers.

In fact take any group G and any element x in G, and consider the
subgroup <x> (the subgroup generated by x). By definition, the group
<x> is called a cyclic group. If <x> is finite (which is true iff x
has finite order in G),<x> is isomorphic to the finite cyclic group
Z_m (the additive group of Z mod m), where m is the order of x, else,
if <x> is infinite, then <x> is isomorphic to the infinite cyclic
group Z (the additive group of integers).

quasi
.

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