Re: English question...
- From: hagman <google@xxxxxxxxxxxxx>
- Date: Sun, 15 Jul 2007 23:23:30 -0700
On 16 Jul., 07:37, "mina_world" <mina_wo...@xxxxxxxxxxx> wrote:
Hello sir~
Show that a group G with at least two elements but with
no proper nontrivial subgroups must be finite and
of prime order.
---------------------------------------------
I want to know what the word "but with" means ?
is these "but" meaningless ?
You may replace "but" with "and"
I.e. show that
|G| > 1 & (H<G => (H=1 v H=G)) => |G|=p
The "&" is the "but"
For reference, I can show it.
Anyway, group G must be cyclic by assumption.
If |G| is infinite, then G~Z.(iso)
so, G has proper nontrivial subgroups.
so, contradiction.
If |G| is not prime,
then there is "d" such that d | |G| and 1< d < |G|.
Let G = <a>
|a^(n/d)| = n / ((n/d), n) = d.
so, <a^(n/d)> is proper nontrivial subgroup.
so, contradiction.
.
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