query: A finite group of even order, whose $2$-Sylow subgroups are cyclic, is not simple.
positive_passages.docid: 36040
positive_passages.text: A Group Having a Cyclic Sylow 2-Subgroup Has a Normal Subgroup.
negative_passages.docid: 22301
negative_passages.text: If $G$ has no proper subgroup, then $G$ is cyclic of prime order
negative_passages.docid: 129830
negative_passages.text: A group with no proper non-trivial subgroups
negative_passages.docid: 53990
negative_passages.text: Group with more than one element and with no proper, nontrivial sub groups must have prime order.
negative_passages.docid: 151025
negative_passages.text: If G is a group of order n=35, then it is cyclic
negative_passages.docid: 28210
negative_passages.text: A finite group with exactly $2$ conjugacy classes isomorphic to $\mathbb{Z}_2$
negative_passages.docid: 145254
negative_passages.text: $G$ be a non-nilpotent and supersoluble finite group.
negative_passages.docid: 31511
negative_passages.text: A finite abelian group that does not contain a subgroup isomorphic to $\mathbb Z_p\oplus\mathbb Z_p$, for any prime $p$, is cyclic.
negative_passages.docid: 54453
negative_passages.text: Finite abelian $p$-group with only one subgroup size $p$ is cyclic
negative_passages.docid: 121810
negative_passages.text: Prove that the product of two infinite cyclic groups is not infinite cyclic.
negative_passages.docid: 58397
negative_passages.text: Every cyclic group with order > 2 has at least 2 distinct generators
negative_passages.docid: 143314
negative_passages.text: Sylow subgroup of $S_{p^2}$
negative_passages.docid: 88718
negative_passages.text: A group is generated by two elements of order $2$ is infinite and non-abelian
negative_passages.docid: 114590
negative_passages.text: Prove that any finite group $G$ of even order contains an element of order $2.$
negative_passages.docid: 50762
negative_passages.text: If $S_n$ has cyclic subgroups of these orders or not?
negative_passages.docid: 124455
negative_passages.text: Why multiplicative group $\mathbb{Z}_n^*$ is not cyclic for $n = 2^k$ and $k \ge 3$
negative_passages.docid: 151619
negative_passages.text: A finite abelian group containing a non-trivial subgroup which lies in every non-trivial subgroup is cyclic