一个二元生成群的元素形式

The Forms of Elements in a Two generated Group

讨论了具有定义关系ｘ－１ｙｘ＝ｙｋ（ｋ∈Ｚ）的二元生成群Ｇ＝＜ｘ，ｙ＞的元素形式．主要结论是Ｇ的元素能表示成形式ｘｍｙｎ（ｍ，ｎ∈Ｚ）当且仅当ｋ＝±１或ｙ的阶有限；当且仅当＜ｙ＞Ｇ．

The present paper deals with the forms of elements in a two generated group G=<x,y> with x -1 yx=y k for some integer k .The main result is that elements of G can be expressed in the form x my n with m,n∈Z,if and only if k=±1 or the order of y (y)<∞; if and only if <y>G.