query: Direct product of groups and isomorphism
positive_passages.docid: 18247
positive_passages.text: Cancellation of Direct Products
negative_passages.docid: 46116
negative_passages.text: Proof of the First Isomorphism Theorem for Groups
negative_passages.docid: 27670
negative_passages.text: Group homomorphisms $\mathbb{Q}\to\mathbb{R}$
negative_passages.docid: 63359
negative_passages.text: Show that the product of all group elements is $e$ or $u$ if $G$ has exactly one involution $u$
negative_passages.docid: 70688
negative_passages.text: How can I prove that the additive group of rationals is not isomorphic to a direct product of two nontrivial groups?
negative_passages.docid: 115663
negative_passages.text: Proving an isomorphism using the isomorphism theorems
negative_passages.docid: 151699
negative_passages.text: Group of inner automorphisms of a group $G$
negative_passages.docid: 3829
negative_passages.text: Number of automorphisms of a direct product of two cyclic $p$-groups
negative_passages.docid: 32428
negative_passages.text: Group homomorphisms and which of the following statements are true (NBHM-$2014$)
negative_passages.docid: 77221
negative_passages.text: Prove that $G$ has a subgroup isomorphic to $G/H$.
negative_passages.docid: 18540
negative_passages.text: Isomorphism from $\mathbb{C}[G]$ to $\prod_{i=1}^h M_{n_i}(\mathbb{C})$.
negative_passages.docid: 11957
negative_passages.text: G is the product of its Sylow subgroups
negative_passages.docid: 24561
negative_passages.text: $\mathbb S_n$ as semidirect product
negative_passages.docid: 50114
negative_passages.text: A question about direct sums and products of modules
negative_passages.docid: 129120
negative_passages.text: Example of a group $G$ such that $G = N_1\cdots N_n$ and $N_i \cap N_j = \{e\}$ for all $i \neq j$ but $G$ is not the internal direct product of them.
negative_passages.docid: 27507
negative_passages.text: Subgroups of Symmetric groups isomorphic to dihedral group
negative_passages.docid: 55778
negative_passages.text: Group theoretic meaning of natural isomorphisms between certain functors