Re: Quotient group question

On 4 Feb, 12:28, quasi <qu...@xxxxxxxx> wrote:
On Mon, 4 Feb 2008 04:25:51 -0800, William Elliot

<ma...@xxxxxxxxxxxxxxxxxx> wrote:
On Mon, 4 Feb 2008, [ISO-8859-1] José Carlos Santos wrote:
On 04-02-2008 10:24, *** wrote:

Here's another question: what is the smallest normal subgroup H of Z*Z
such that (Z*Z)/H is isomorphic to Z?

What about answering first to the questions raised by your previous
question?

He tacidly addressed them with his revised question.
There is however, no smallest normal subgroup of Z^2.

The group in question is Z*Z, not Z^2.

quasi

Yes, but if we are restricting our attention to the study of normal
subgroups H of Z*Z, or alternatively of Z^2, such that Z*Z/H, or
respectively Z^2/H, is isomorphic to Z, then there is in some sense no
difference, because all subgroups H of Z*Z with that property contain
the commutator subgroup, C say, of Z*Z, and Z*Z/C is isomorphic to
Z^2.

Of course, as you pointed out, the answer to both questions is that
there is no unique smallest subgroup with this property.

Derek Holt.
.

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