4
$\begingroup$

I am reading a first course in abstract algebra and there is a claim that says a group $G$ with no proper non trivial subgroups is cyclic. But I don't understand what does it mean to have no proper non trivial subgroup. I know that the identity element is trivial subgroup, all other subgroups are nontrivial and $G$ is the improper subgroup of $G$, and all others are proper subgroups. But what is proper non trivial subgroup? Thanks

$\endgroup$
  • $\begingroup$ I think one should read "a group G which has not proper non trivial subgroups is cyclic" $\endgroup$ – Boris Novikov Mar 15 '13 at 23:09
  • 1
    $\begingroup$ A "proper nontrivial subgroup" is a subgroup which is both a proper subgroup and a nontrivial subgroup. $\endgroup$ – Marcel Besixdouze Mar 18 '15 at 3:15
3
$\begingroup$

For example, in $\{0,1,2,3\}$ (cyclic group of order $4$) the elements $\{0,2\}$ make a subgroup. This is a nontrivial subgroup, and it is not the entire group, so it is a proper subgroup.

The point is that a subgroup is also a subset. Subsets can be proper, or improper (i.e. equal to the big set). Proper subgroups are proper subsets.

$\endgroup$
  • $\begingroup$ Funny we came up with the same example of proper subgroup. :) $\endgroup$ – A.P. Mar 15 '13 at 23:03
  • $\begingroup$ I interpreted no proper nontrivial as no proper and no non trivial, but it means no (proper non trivial). Thanks $\endgroup$ – Yasin Razlık Mar 15 '13 at 23:05
  • 1
    $\begingroup$ @A.P.: It's the first one. Literally! $\endgroup$ – Asaf Karagila Mar 15 '13 at 23:07
  • $\begingroup$ @bigO: No problem! $\endgroup$ – Asaf Karagila Mar 15 '13 at 23:12
4
$\begingroup$

A subgroup N of a group $G$ is said to be proper if $N\neq G$ and to be non-trivial if $N\neq \{e\}$, where $e$ is the identity of $G$.

For example $N=\{0,2\}$ is a proper subgroup of $(\Bbb Z/4\Bbb Z,+)$, isomorphic to $\Bbb Z/2\Bbb Z$.

$\endgroup$
  • $\begingroup$ I think my brain freezed for a moment. I see now thanks $\endgroup$ – Yasin Razlık Mar 15 '13 at 23:05
1
$\begingroup$

If a nontrivial group has no proper nontrivial subgroup, then it is a cyclic group of prime order. In other words, it is generated by a single element whose order is a prime number.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.