query: $G$ is a finite group. $A$ and $B$ are non-empty subsets of $G$. If $\vert A\vert+\vert B\vert>\vert G\vert$, prove that $AB=G$.
positive_passages.docid: 7121
positive_passages.text: If $G$ is a finite group and $|G| < |A| + |B|$, then $G=AB$.
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: 140334
negative_passages.text: Let $G$ be a group and $H$ be a subgroup of $G$ of index $2$ .Then show that if $a,b\notin H$ then $ab\in H$.
negative_passages.docid: 33233
negative_passages.text: Need to prove that $(S,\cdot)$ defined by the binary operation $a\cdot b = a+b+ab$ is an abelian group on $S = \Bbb R \setminus \{-1\}$.
negative_passages.docid: 58205
negative_passages.text: $ N $ normal in a finite group $ G $, $ |N| = 5 $ and $ |G| $ odd. Why is $ N \subseteq Z(G) $?
negative_passages.docid: 35800
negative_passages.text: If $K \leq H \leq G$, show that $[G:K] = [G:H][H:K]$.
negative_passages.docid: 45760
negative_passages.text: if G is generated by {a,b} and ab=ba, then prove G is Abelian
negative_passages.docid: 43759
negative_passages.text: Prove that the set $\mathrm{Aut}(G)$ of all automorphisms of the group $G$ with the operation of taking the composition is a group
negative_passages.docid: 21217
negative_passages.text: $G$, group. $a,b\in G$. Show that $|a|=|a^{-1}|; |ab|=|ba|$, and $|a|=|cac^{-1}|, \forall c\in G$.
negative_passages.docid: 2761
negative_passages.text: If $S$ is a nonempty subset of group $G$, then $S^{|G|}$ is a subgroup of $G$.
negative_passages.docid: 13834
negative_passages.text: Show that G is a group, if G is finite, the operation is associative, and cancellation law holds
negative_passages.docid: 77221
negative_passages.text: Prove that $G$ has a subgroup isomorphic to $G/H$.
negative_passages.docid: 10348
negative_passages.text: Prove that if the $\gcd(a,b)=1$, then $\gcd(a+b,ab)=1$
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: 145406
negative_passages.text: Prove every element of $G$ has finite order.
negative_passages.docid: 32735
negative_passages.text: If $ \langle G, \star \rangle $ is an abelian group, then for all $a, b \in G$, show that $(a \star b)^{n} = a^{n} \star b^{n}$.
negative_passages.docid: 142228
negative_passages.text: Proof that a subset closed under group operation of a finite group is a subgroup