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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

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I think one should read "a group G which has not proper non trivial subgroups is cyclic" – Boris Novikov Mar 15 '13 at 23:09
A "proper nontrivial subgroup" is a subgroup which is both a proper subgroup and a nontrivial subgroup. – Marcel T. Mar 18 '15 at 3:15
up vote 1 down vote accepted

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.

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

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 $\Bbb Z/2\Bbb Z$.

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I think my brain freezed for a moment. I see now thanks – bigO Mar 15 '13 at 23:05

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.

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