query: Let $G$ be a group and let $a, b$ be elements of $G$, show that $|ab|=l.c.m (|a|,|b|)$
positive_passages.docid: 148108
positive_passages.text: Let $L=\operatorname{lcm}(o(a), o(b))$ and prove that $(ab)^L =e$.
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: 85011
negative_passages.text: Prove that $\exists a,g,h\in G\colon h=aga^{-1}, g\neq h ,gh=hg$ in a finite non-abelian group $G$.
negative_passages.docid: 154398
negative_passages.text: Show that H is a subgroup of G
negative_passages.docid: 63359
negative_passages.text: Show that the product of all group elements is $e$ or $u$ if $G$ has exactly one involution $u$
negative_passages.docid: 32253
negative_passages.text: How to prove $b=c$ if $ab=ac$ (cancellation law in groups)?
negative_passages.docid: 112969
negative_passages.text: Show that $ab + bc + ac \ge a\sqrt{bc} + b\sqrt{ac} + c\sqrt{ab}$
negative_passages.docid: 143550
negative_passages.text: Order of product of two elements in an abelian group
negative_passages.docid: 90332
negative_passages.text: Prove that that $U(n)$ is an abelian group.
negative_passages.docid: 79452
negative_passages.text: How to prove that it is a group?
negative_passages.docid: 5297
negative_passages.text: If G is a finite abelian group and $a_1,...,a_n$ are all its elements, show that $x=a_1a_2a_3...a_n$must satisfy $x^2=e$.
negative_passages.docid: 114221
negative_passages.text: A Group $G$ is soluble $\iff G^{(k)}=\{e\}$ for some $k$
negative_passages.docid: 71420
negative_passages.text: Prove that a group is abelian.
negative_passages.docid: 105471
negative_passages.text: How to prove that if each element of group is inverse to itself then group commutative?
negative_passages.docid: 120642
negative_passages.text: Let $g : \Bbb N \times \Bbb N \to\Bbb N \times \Bbb N$ defined as $g(m,n) = (m + n,m - n)$
negative_passages.docid: 117701
negative_passages.text: Let $G$ be a group and $X$ set. $a,b \in X$ are in the same orbit, so show that $stab(a) \cong stab (b)$
negative_passages.docid: 26624
negative_passages.text: Prove that is group is abelian