Two cosets of the same subgroup either coincide or do not meet.Suggest a slogan
Let \(G\) be a group, \(H\leq G\) a subgroup, and \(x,y\in G\). Then the left cosets \(xH\) and \(yH\) are either equal or disjoint.
Suppose \(g\in xH\cap yH\), say \(g=xh=yh'\) with \(h,h'\in H\). Then \(x=yh'h^{-1}\in yH\), so \(xH\subseteq yHH=yH\); by symmetry \(yH\subseteq xH\).