hm:new.gen\], noting that $H\lhd G$. Let $\pi:G\to G/H$ be the quotient homomorphism, and note that $$\begin{aligned}
\frac{|\pi(AP)|}{|\pi(P)|}&=\frac{|APH|}{|PH|}\\
   &\le\exp(e^{O(s^2)}\log^{O(s)}2K)\frac{|APH|}{|AH|}\\
   &=\exp(e^{O(s^2)}\log^{O(s)}2K)\frac{|\pi(AP)|}{|\pi(A)|}\\
   &\le\exp(e^{O(s^2)}\log^{O(s)}2K),\end{aligned}$$ the last inequality coming from the fact that $\pi(A)$ is a $K$-approximate group and $\pi(AP)\subset\pi(A)^{e^{O(s^2)}\log^{O(s)}2K}$. Applying \[lem:covering\] in the quotient $G/H$ therefore gives a set $X\subset A$ of size at most $\exp(e^{O(s^2)}\log^{O(s)}K)$ such that $A\subset XHPP^{-1}$. Now $PP^{-1}\subset A^{e^{O(s^2)}\log^{O(s)}2K}$ is an ordered progression of rank double that of $P$, which is still at most $e^{O(s^2)}\log^{O(s)}2K$. The corollary therefore follows from \[prop:nilprog.equiv\].

We may assume that $A$ generates $G$. Let $H$ and $P_0=P_{\text{\textup{ord}}}(x;L)$ be as given by \[thm:new.gen\], noting that $H\lhd G$. Let $\pi:G\to G/H$ be the quotient homomorphism, noting that $$\frac{|\pi(P_0)|}{|\pi(A)|}=\frac{|P_0H|}{|AH|}\ge\exp(-e^{O(s^2)}\log^{O(s)}2K).$$ Applying and \[lem:chang\] in the quotient $G/H$, we therefore have $$t\le e^{O(s^2)}\log^{O(s)}2K$$ and sets $S_1,\ldots,S_t\subset A$ with $|S_i|\le2K$ such that $$A\subset S_{t-1}^{-1}\cdots S_1^{-1}P_0^{-1}P_0S_1\cdots S_tH.$$ Enumerating the elements of each $S_i$ as $s_{1,i},\ldots,s_{r_i,i}$ and writing $$Q_i=\{s_{1,i}^{\epsilon_1}\cdots s_{r_i,i}^{\epsilon_{r_i}}:\epsilon_j,\in\{-1,0,1\}\},$$ the set $P=Q_{t-1}\cdots Q_1P_0^{-1}P_0Q_1\cdots Q_t$ is therefore an ordered progression of rank at most $$e^{O(s^2)}K\log^{O(s)}2K$$ satisfying $$A\subset PH\subset A^{4Kt+e^{O(s^2)}\log^{O(s)}2K}H\subset A^{e^{O(s^2)}K\log^{O(s)}2K}H.$$ The corollary therefore follows from \[prop:nilprog.equiv\].

Applications to non-nilpotent groups {#sec:non-nilp}
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In this section we use our results to improve the bounds on the ranks of the coset nilprogressions appearing in