query: Prove H = {a ∈ G | aga^-1 =g} ≤ G
positive_passages.docid: 146760
positive_passages.text: Prove that H is a subgroup of G
negative_passages.docid: 137707
negative_passages.text: How to prove that $(A\cup B) \cap C \subseteq A\cup (B\cap C)$
negative_passages.docid: 94571
negative_passages.text: How can we prove that $N_G(H)=H$ implies $N_{f(G)}(f(H))=f(H)$
negative_passages.docid: 93596
negative_passages.text: If $H\leq G$ and $[G:H]=2$, then $H$ is normal in $G$.
negative_passages.docid: 35800
negative_passages.text: If $K \leq H \leq G$, show that $[G:K] = [G:H][H:K]$.
negative_passages.docid: 34567
negative_passages.text: If $G_1$ and $G_2$ are finite group and $H\leq G_1\times G_2$ then $H=P_1\times P_2$ with $P_i\leq G_i$?
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: 31331
negative_passages.text: $|a|=m,\,\gcd(m,n)=1 \implies a$ is an $n$'th power
negative_passages.docid: 81618
negative_passages.text: Prove $||a|-|b| | \leq |a-b|$
negative_passages.docid: 151852
negative_passages.text: How to prove $A \subset B \iff A \cup B = B$
negative_passages.docid: 56650
negative_passages.text: Prove that $(G, \circ)$ is a group if $a\circ x = b$ and $x\circ a = b$ have unique solutions
negative_passages.docid: 29996
negative_passages.text: Prove that if a normal subgroup $H$ of $ G$ has index $n$, then $g^n \in H$ for all $g \in G$
negative_passages.docid: 68705
negative_passages.text: Prove that $A\ge0, B\ge0$ and $A\ge B$ implies $B^{-1}\ge A^{-1}$
negative_passages.docid: 58033
negative_passages.text: $G\times H\cong G$ with $H$ non-trivial
negative_passages.docid: 150585
negative_passages.text: Group with $a=a^{-1}$ for all $a\in G$ is abelian
negative_passages.docid: 44729
negative_passages.text: Prove $g$ is a generator if $g^q=1 \pmod p$
negative_passages.docid: 130455
negative_passages.text: If $H,K⊲G$ and $H∩K = \{1_G\}$, then all elements in $H$ commute with all elements in $K$