query: Group theory - isomorphism and cyclic groups
positive_passages.docid: 146969
positive_passages.text: Proving that every group of order $4$ is isomorphic to $\Bbb Z_4$ or $\Bbb Z^*_8$
negative_passages.docid: 152071
negative_passages.text: Group $G$ is cyclic $\iff$ every subgroup of $G$ has the form $G^k$, where $G^k=\{g^k\mid g\in G\}$
negative_passages.docid: 33455
negative_passages.text: Question on group homomorphisms involving the standard Z-basis
negative_passages.docid: 38153
negative_passages.text: Looking for a simple proof that groups of order $2p$ are up to isomorphism $\mathbb{Z}_{2p}$ and $D_{p}$ .
negative_passages.docid: 157040
negative_passages.text: For groups $A,B,C$, if $A\times B$ and $A\times C$ are isomorphic do we have $B$ isomorphic to $C$?
negative_passages.docid: 107687
negative_passages.text: How to prove that $U_{2^n}$ is isomorphic as group to $\mathbb Z_2 \times \mathbb Z_{2^{n-2}}$ for $n \ge 3$?
negative_passages.docid: 158074
negative_passages.text: Cyclic group $G$ with generator $ab$.
negative_passages.docid: 93169
negative_passages.text: Show that $(\mathbb Z[x],+)$ and $(\mathbb Q_{>0},\cdot)$ are isomorphic groups
negative_passages.docid: 126098
negative_passages.text: Isomorphism of factor groups
negative_passages.docid: 39938
negative_passages.text: Subgroups of a cyclic group and their order.
negative_passages.docid: 76504
negative_passages.text: Non-trivial cyclic group with unique generator
negative_passages.docid: 110808
negative_passages.text: Isomorphisms preserve cyclic
negative_passages.docid: 101602
negative_passages.text: consider the group $G=\mathbb Q/\mathbb Z$. For $n>0$, is there a cyclic subgroup of order n
negative_passages.docid: 152603
negative_passages.text: Group isomorphic to its automorphism group
negative_passages.docid: 94239
negative_passages.text: Proof that if group $G/Z(G)$ is cyclic, then $G$ is commutative
negative_passages.docid: 32428
negative_passages.text: Group homomorphisms and which of the following statements are true (NBHM-$2014$)
negative_passages.docid: 2254
negative_passages.text: Finite group with isomorphic normal subgroups and non-isomorphic quotients?