query: Proof that every group of order 30 has a subgroup or order 15
positive_passages.docid: 31687
positive_passages.text: Classification of groups of order 30
negative_passages.docid: 51359
negative_passages.text: Prove that every subgroup of an abelian group is a normal subgroup.
negative_passages.docid: 133298
negative_passages.text: Assume $|G|=40$. Show that the subgroup of order $8$ is normal and unique.
negative_passages.docid: 129568
negative_passages.text: How many different groups of order $15$ there are?
negative_passages.docid: 37913
negative_passages.text: Prove that $ AB$ is a subgroup of order $mn$ if $m$ and $n$ are relatively prime.
negative_passages.docid: 12667
negative_passages.text: How many subgroups of order $n$ does $D_n$ have?
negative_passages.docid: 148774
negative_passages.text: show elements of order 15 in $S_8$ satisfy a relation
negative_passages.docid: 8841
negative_passages.text: A $p$-group of order $p^n$ has a normal subgroup of order $p^k$ for each $0\le k \le n$
negative_passages.docid: 58863
negative_passages.text: Proof that $A_n$ the only subgroup of $ S_n$ index $2$.
negative_passages.docid: 76943
negative_passages.text: Existence of subgroup of order six in $A_4$
negative_passages.docid: 149578
negative_passages.text: Subgroups of $(\mathbb Z_n,+)$
negative_passages.docid: 126113
negative_passages.text: Prove the existence of order 4 subgroups of order 8 groups
negative_passages.docid: 117899
negative_passages.text: At most one subgroup of every order dividing $\lvert G\rvert$ implies $G$ cyclic
negative_passages.docid: 16259
negative_passages.text: How many non isomorphic groups of order 30 are there?
negative_passages.docid: 131516
negative_passages.text: Subgroup generated by n-th powers of elements
negative_passages.docid: 129937
negative_passages.text: How to prove that $A_5$ has no subgroup of order 30?
negative_passages.docid: 105139
negative_passages.text: Verification of Proof that a nonabelian group G of order pq where p and q are primes has a trivial center