Subgroup indices multiply in towers.Suggest a slogan
Let \(G\) be a group and \(K\leq H \leq G\) subgroups. Then \[ [G: K] = [G:H]\, [H: K] \] as cardinals; in particular \([G:K]\) is finite if and only if \([G:H]\) and \([H:K]\) are both finite.
Write \(G=\coprod_{i\in I}g_iH\) and \(H=\coprod_{j\in J}h_jK\) as disjoint unions of left cosets, with \(\abs{I}=[G:H]\) and \(\abs{J}=[H:K]\). Then \(G=\bigcup_{i,j}g_ih_jK\). If \(g_ih_jK=g_{i'}h_{j'}K\), then \(g_ih_j\in g_{i'}h_{j'}K\subseteq g_{i'}H\), so \(g_iH=g_{i'}H\) and \(i=i'\); cancelling \(g_i\) gives \(h_jK=h_{j'}K\) and \(j=j'\). Hence the cosets \(g_ih_jK\) are pairwise distinct, and \([G:K]=\abs{I\times J}\).