English question...

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.

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I want to know what the word "but with" means ?
is these "but" meaningless ?

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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Relevant Pages

  • Re: English question...
    ... of prime order. ... If |G| is infinite, then G~Z. ... G has proper nontrivial subgroups. ... so, contradiction. ...
    (sci.math)
  • Re: English question...
    ... However 'but' denotes some contrast. ... must be finite and of prime order. ...
    (sci.math)
  • Re: English question...
    ... However 'but' denotes some contrast. ... I'd translate this usage as ... must be finite and of prime order. ... G has proper nontrivial subgroups. ...
    (sci.math)