query: $Z(G\ast H)=\{e\}$ and $G\ast H$ is an infinite group
positive_passages.docid: 40451
positive_passages.text: Center of a free product
negative_passages.docid: 98739
negative_passages.text: If $H$ is a subgroup of $G$ and $K$ is a subgroup of $H$, then $$|G:K|=|G:H||H:K|$$
negative_passages.docid: 5226
negative_passages.text: Prove that a finite semigroup G with identity is a group if and only if G contains only one idempotent
negative_passages.docid: 113708
negative_passages.text: Correspondence between Ext group and extensions (from Weibel's book)
negative_passages.docid: 77221
negative_passages.text: Prove that $G$ has a subgroup isomorphic to $G/H$.
negative_passages.docid: 11000
negative_passages.text: Fundamental group of $\mathbb{R}^2 \setminus (\mathbb{Z} \times \{0\})$
negative_passages.docid: 97565
negative_passages.text: Quotient group $G/H$ is abelian iff $[G,G] \subseteq H$
negative_passages.docid: 45310
negative_passages.text: If $H\leq Z (G)$ and $G/H$ is nilpotent, then $G$ is nilpotent.
negative_passages.docid: 108106
negative_passages.text: If M is a non-orientable closed connected 3 manifold prove H1(M) is an infinite group.
negative_passages.docid: 23579
negative_passages.text: If $G$ is isomorphic to $H$, then ${\rm Aut}(G)$ is isomorphic to ${\rm Aut}(H)$
negative_passages.docid: 64193
negative_passages.text: Cardinality of Infinite Symmetric group
negative_passages.docid: 87882
negative_passages.text: If $N \lhd G$ is abelian and $G/N$ is abelian then $G$ is abelian
negative_passages.docid: 106277
negative_passages.text: Prove that $G$ contains a subgroup $H_i$ for each $i\in\Bbb Z$ with $\dots\subset H_{−2}\subset H_{−1}\subset H_0\subset H_1\subset\dots?$
negative_passages.docid: 5089
negative_passages.text: An element of a group has the same order as its inverse
negative_passages.docid: 154398
negative_passages.text: Show that H is a subgroup of G
negative_passages.docid: 135188
negative_passages.text: Show that if $\vert G\vert = pq$, then either $G$ is abelian or $Z(G) = \{ e\}$.
negative_passages.docid: 4838
negative_passages.text: For $N\unlhd G$ , with $C_G(N)\subset N$ we have $G/N$ is abelian