I have a question about a reduction argument from Jarden's and Lubotzky's paper ['Elementary equivalence of profinite groups'][1] in Lemma 1.1 on page 3: *Lemma 1.1: For each positive integer $n$ and each finite group $A$ of order at most $n$ there exists a sentence $ \theta $ of $\mathcal{L} \text{(group)} $ such that for every group $G$ of order at most $n$ the sentence $ \theta $ holds in $G$ if and only if $A$ is a quotient of $G$.* $\tag{L}$ The proof begins with a reduction step I not understand: *Proof*: It suffices to prove that for every positive integer $n$ and for every group $A$ of order $d$ dividing $n$ there exists a sentence $ \theta $ of $\mathcal{L} \text{(group)} $ with the following property: for every group $G$ of order $n$ the sentence $ \theta $ holds in $G$ if and only if $G$ has a normal subgroup $M$ such that $G/M \cong A$. [..] $\tag{L'}$ Assume we have proved (L'). Why the claim (L) of the lemma follows immediately from (L'); in other words why is sufficient to show only claim (L'), which is seemingly weaker as the claim (L)? I have already asked identical question in [MathStackEx][2] but a comment by Noah Schweber stresses another important viewpoint: That this lemma 1.1 is an elementary consequence of a much stronger statement about structures in context of finite languages. That's true, but that was not my original concern: my original concern is just about the logic of the proof itself: why (L') *implies* (L). [1]: http://132.64.72.10/~alexlub/PAPERS/Elementary%20equivalence%20of%20profinite%20groups/Elementary%20equivalence%20of%20profinite%20groups.pdf [2]: http://%20https://math.stackexchange.com/questions/3911907/lemma-from-paper-on-elementary-equivalence-of-profinite-groups-by-jarden-lubot