query: If ${\rm Aut}(G) = 1$ , show that order of $G$ is $1$ or $2.$
positive_passages.docid: 52121
positive_passages.text: Let $G$ be a finite group with $|G|>2$. Prove that ${\rm Aut}(G)$ contains at least two elements.
negative_passages.docid: 8742
negative_passages.text: Examples and further results about the order of the product of two elements in a group
negative_passages.docid: 8434
negative_passages.text: Order of $\mathbb{Z}[i]/(1+i)$
negative_passages.docid: 13807
negative_passages.text: Is there a way to prove that the order of an element in a Group divides the order of the Group, WITHOUT USING LAGRANGE'S
negative_passages.docid: 15104
negative_passages.text: Show that $\big|\frac{a-b}{1-\overline{a}b}\big|=1$, when either $|a|=1$ or $|b|=1$.
negative_passages.docid: 58654
negative_passages.text: If $|A| > \frac{|G|}{2} $ then $AA = G $
negative_passages.docid: 151559
negative_passages.text: Please help: How to show that Aut(G) is a subgroup of Sym(G)
negative_passages.docid: 7151
negative_passages.text: Let $a,m,n \in \mathbf{N}$. Show that if $\gcd(m,n)=1$, then $\gcd(a,mn)=\gcd(a,m)\cdot\gcd(a,n)$.
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: 35800
negative_passages.text: If $K \leq H \leq G$, show that $[G:K] = [G:H][H:K]$.
negative_passages.docid: 14383
negative_passages.text: Order of Permutation : If $\tau \in S_n$ has order $m$, then $\sigma \tau\sigma^{-1}$ has also order $m$.
negative_passages.docid: 36887
negative_passages.text: Suppose $G$ is a group of order 4. Show either $G$ is cyclic or $x^2=e$.
negative_passages.docid: 119758
negative_passages.text: On the order of elements of $\mathrm{GL}(2,q)$.
negative_passages.docid: 83178
negative_passages.text: Matrix Group induction proof and order of elements question
negative_passages.docid: 16476
negative_passages.text: Order of the product of two commuting elements with coprime orders in a group.
negative_passages.docid: 119808
negative_passages.text: Proving that $n$-cycle has order $n$
negative_passages.docid: 80148
negative_passages.text: Prove $\gcd(a+b,a^2+b^2)$ is $1$ or $2$ if $\gcd(a,b) = 1$