query: Automorphism group of direct product of groups, proof
positive_passages.docid: 146278
positive_passages.text: Automorphism group of direct product of groups
negative_passages.docid: 147298
negative_passages.text: How to find all subgroups of a direct product?
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: 129610
negative_passages.text: Why is the direct product of a finite number of nilpotent groups nilpotent?
negative_passages.docid: 137739
negative_passages.text: Prove that the group isomorphism $\mathbb{Z}^m \cong \mathbb{Z}^n$ implies that $m = n$
negative_passages.docid: 42096
negative_passages.text: Direct product of Sylow subgroups
negative_passages.docid: 32428
negative_passages.text: Group homomorphisms and which of the following statements are true (NBHM-$2014$)
negative_passages.docid: 66863
negative_passages.text: How is the quotient group related to the direct product group?
negative_passages.docid: 94239
negative_passages.text: Proof that if group $G/Z(G)$ is cyclic, then $G$ is commutative
negative_passages.docid: 114811
negative_passages.text: Characteristic subgroups and automorphisms
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: 55393
negative_passages.text: Existence of a group isomorphism between $(\mathbb K,+)$ and $(\mathbb K^\times,\cdot)$
negative_passages.docid: 13761
negative_passages.text: Automorphisms in $\mathbb{Z}_n$
negative_passages.docid: 70394
negative_passages.text: Prove that $a * b$ is a group (Hungerford, 2nd ed., sec $7.2$, Exercise $22$)
negative_passages.docid: 46260
negative_passages.text: Prove that a group generated by two elements of order $2$, $x$ and $y$, is isomorphic to $D_{2n}$, where $n = |xy|.$
negative_passages.docid: 158490
negative_passages.text: Prove that $G^{op} \cong G$ when $G^{op}$ is the opposite group of G
negative_passages.docid: 90511
negative_passages.text: Commutator subgroup of a dihedral group.