query: let $x$ be in finite group $G$ and let order of $x$ is $p$. If $h^{-1}xh = x^{10}$ for a finite group show that $p=3$
positive_passages.docid: 95229
positive_passages.text: Qualifying Exam Question On Elementary Group Theory
negative_passages.docid: 90449
negative_passages.text: Showing that a Sylow subgroup of a group of order $p^{2}q$ is normal.
negative_passages.docid: 39165
negative_passages.text: Suppose $G$ is a group and $H,K \trianglelefteq G$. If $H \cap K =1$ the $hk=kh$ for all $k \in K, h \in H$?
negative_passages.docid: 106577
negative_passages.text: If G is not abelian with order $p^3$ then $G'=Z(G)$
negative_passages.docid: 119512
negative_passages.text: If H ≤ Z(G) ≤ G, where G is a finite group,Z(G) is its center, and (G:H) = p for some prime p, then G is abelian.
negative_passages.docid: 154398
negative_passages.text: Show that H is a subgroup of G
negative_passages.docid: 107935
negative_passages.text: If $G$ is a group, $H$ is a subgroup of $G$ and $g\in G$, is it possible that $gHg^{-1} \subset H$?
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: 121587
negative_passages.text: Let $H$ be a normal subgroup of index $n$ in a group $G$. Show that for all $g \in G, g^n \in H$
negative_passages.docid: 56607
negative_passages.text: gcd and order of elements of group
negative_passages.docid: 17178
negative_passages.text: Order of $\phi(g)$ divides the order of $g$
negative_passages.docid: 121423
negative_passages.text: Prove that every element of a finite group has an order
negative_passages.docid: 83273
negative_passages.text: Show that $o(gHg^{-1})=o(H)$
negative_passages.docid: 85579
negative_passages.text: $G$ a group s.t. every non-identity element has order 2. If $G$ is finite, prove $|G| = 2^n$ and $G \simeq C_2 \times C_2 \times\cdots\times C_2$
negative_passages.docid: 15387
negative_passages.text: Order of $GL(n, \mathbb Z_p)$
negative_passages.docid: 83654
negative_passages.text: Order of an element in a finite cyclic group
negative_passages.docid: 79995
negative_passages.text: G is group of order pq, pq are primes