[proofplan] This entry records the published Breuillard-Green-Tao classification theorem for finite approximate groups, using the standard definitions of approximate group, control, and coset nilprogression. To avoid circularity, the external input is the original Breuillard-[Green-Tao theorem](/theorems/4614) as a literature theorem, not this Androma entry itself. We first unpack the definitions needed to compare hypotheses and conclusions, then quote the published theorem in its quantitative form, and finally identify its conclusion with the statement here. [/proofplan] [step:Unpack the approximate group and control definitions] Let $1_G \in G$ denote the identity element of the group $G$. Since $A \subset G$ is a finite [$K$-approximate group](/page/Approximate%20Group), by definition $A$ is finite, $A = A^{-1}$, $1_G \in A$, and there exists a finite set $X \subset G$ satisfying \begin{align*} |X| &\leq K, & A \cdot A &\subset X \cdot A. \end{align*} For a parameter $M \geq 1$, say that a subset $B \subset G$ $M$-controls $A$ if $|B| \leq M|A|$ and there exists a set $Y \subset G$ with $|Y| \leq M$ such that \begin{align*} A &\subset Y \cdot B \cap B \cdot Y. \end{align*} A [coset nilprogression](/page/Coset%20Nilprogression) of rank at most $r$ and step at most $s$ means a finite set $P \subset G$ obtained as the preimage of a nilprogression of rank at most $r$ and step at most $s$ under the quotient map from a subgroup of $G$ by a finite [normal subgroup](/page/Normal%20Subgroup). These are the notions of approximate group, control, and coset nilprogression used in the Breuillard-Green-Tao structure theorem. [guided] We first expand the definitions so that the quoted theorem can be compared with the statement without hidden terminology. The ambient object is a [group](/page/Group) $G$, and the subset under consideration is $A \subset G$. The phrase finite [$K$-approximate group](/page/Approximate%20Group) means the following four concrete conditions hold: $A$ is finite, $A$ is symmetric in the sense that $A = A^{-1}$, the identity element $1_G$ belongs to $A$, and there is a finite set $X \subset G$ with \begin{align*} |X| &\leq K, & A \cdot A &\subset X \cdot A. \end{align*} We also record the conclusion terminology. For $M \geq 1$, a subset $B \subset G$ $M$-controls $A$ if $|B| \leq M|A|$ and there is a set $Y \subset G$ with $|Y| \leq M$ such that \begin{align*} A &\subset Y \cdot B \cap B \cdot Y. \end{align*} A [coset nilprogression](/page/Coset%20Nilprogression) of rank at most $r$ and step at most $s$ is a finite subset $P \subset G$ obtained as the preimage of a nilprogression of rank at most $r$ and step at most $s$ under the quotient map from a subgroup of $G$ by a finite normal subgroup. These definitions are exactly the definitions used in the published Breuillard-Green-Tao classification theorem, so the comparison of hypotheses and conclusions is now a comparison of explicit statements. [/guided] [/step] [step:Quote the published Breuillard-Green-Tao classification theorem] We use the following external result of Breuillard, Green, and Tao: for every $K \geq 1$ there exist constants $C(K) > 0$, $r(K) \in \mathbb{N}$, and $s(K) \in \mathbb{N}$ such that, whenever $G_0$ is an arbitrary group and $B \subset G_0$ is a finite $K$-approximate group, there is a coset nilprogression $P \subset G_0$ of rank at most $r(K)$ and step at most $s(K)$ which $C(K)$-controls $B$. This is the published Breuillard-Green-Tao approximate group structure theorem, specifically Theorem 2.10 of Breuillard-Green-Tao, "The structure of approximate groups", Publications mathématiques de l'IHÉS 116 (2012), 115-221. Applying this literature theorem with $G_0 := G$ and $B := A$ is valid by the hypotheses unpacked in the preceding step. Hence there exist constants $C(K) > 0$, $r(K) \in \mathbb{N}$, and $s(K) \in \mathbb{N}$ depending only on $K$, and a coset nilprogression $P \subset G$ of rank at most $r(K)$ and step at most $s(K)$, such that $P$ $C(K)$-controls $A$. [guided] Now we state the external input precisely, because this is the only non-formal ingredient in the proof. The published Breuillard-Green-Tao approximate group structure theorem says: for every $K \geq 1$ there are constants $C(K) > 0$, $r(K) \in \mathbb{N}$, and $s(K) \in \mathbb{N}$ such that, for every arbitrary group $G_0$ and every finite $K$-approximate group $B \subset G_0$, there exists a coset nilprogression $P \subset G_0$ of rank at most $r(K)$ and step at most $s(K)$ which $C(K)$-controls $B$. The reference is Breuillard-Green-Tao, "The structure of approximate groups", Publications mathématiques de l'IHÉS 116 (2012), Theorem 2.10. This invocation is not circular: the external theorem is the published classification theorem in the literature, while the present Androma theorem is recording that result in the site's terminology. We verify the hypotheses for the substitution $G_0 := G$ and $B := A$. The statement assumes that $G$ is a group and $A \subset G$ is a finite $K$-approximate group, and the preceding step expanded that assumption into the exact finiteness, symmetry, identity, and covering conditions required by the published theorem. Therefore the theorem applies and produces constants $C(K) > 0$, $r(K) \in \mathbb{N}$, and $s(K) \in \mathbb{N}$ depending only on $K$, together with a coset nilprogression $P \subset G$ of rank at most $r(K)$ and step at most $s(K)$ which $C(K)$-controls $A$. [/guided] [/step] [step:Identify the obtained control statement with the theorem statement] The preceding step gives a coset nilprogression $P \subset G$ whose rank and step are bounded by the functions $r(K)$ and $s(K)$, and it gives $C(K)$-control of $A$ by $P$. Since the only parameters appearing in $C(K)$, $r(K)$, and $s(K)$ are functions of $K$, this is exactly the assertion that $A$ is controlled, with constants depending only on $K$, by a coset nilprogression of rank and step depending only on $K$. [/step]