query: Order of Homomorphic image of an element of a group divides the order of that element.
positive_passages.docid: 22995
positive_passages.text: Let f:G to H be 1-1 homomorphism show that order of f(a) divides the order of G.
negative_passages.docid: 129750
negative_passages.text: Order of $SL_2(\mathbb{Z}/ n\mathbb{Z})$
negative_passages.docid: 38853
negative_passages.text: Cyclic group of prime order
negative_passages.docid: 933
negative_passages.text: How many groups exist with order $n$ (two isomorphic groups are treated as the same group)
negative_passages.docid: 98734
negative_passages.text: Order of a product of elements in a generic group
negative_passages.docid: 28676
negative_passages.text: Is a finite group uniquely determined by the orders of its elements?
negative_passages.docid: 54911
negative_passages.text: The preimage of a normal subgroup under a group homomorphism is normal
negative_passages.docid: 124309
negative_passages.text: $g\in G$ maximal order in $G$ abelian then $G=\left\oplus H$
negative_passages.docid: 38083
negative_passages.text: Order of conjugate of an element given the order of its conjugate
negative_passages.docid: 143550
negative_passages.text: Order of product of two elements in an abelian group
negative_passages.docid: 114187
negative_passages.text: In a finite cyclic group, the generator has the order of the group
negative_passages.docid: 21568
negative_passages.text: Maximal order of an element in a symmetric group
negative_passages.docid: 37061
negative_passages.text: The order of $ab$ when $a,b$ commute
negative_passages.docid: 39938
negative_passages.text: Subgroups of a cyclic group and their order.
negative_passages.docid: 27728
negative_passages.text: Question on a homomorphism of a set G.
negative_passages.docid: 55719
negative_passages.text: Order of general- and special linear groups over finite fields.
negative_passages.docid: 8841
negative_passages.text: A $p$-group of order $p^n$ has a normal subgroup of order $p^k$ for each $0\le k \le n$