A nonempty subset $D$ of a group $G$ is called


$\bullet$ *decomposable* if $D\subseteq DD$, that is every element $x\in D$ is can be written as the product $x=yz$ of some elements $y,z\in D$;


$\bullet$ *product-one* if there exists $n\in\mathbb N$ and pairwise distinct elements $x\_1,\dots,x\_n\in D$ such that $x\_1\cdots x\_n=1$.



> 
> **Problem 1.** Let $D$ be a finite decomposable subset of a group. Is $D$ product-one?
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> 
> 


**Remark 1.** For commutative groups this problem was posed by [Gjergji Zaimi](https://mathoverflow.net/q/16857/61536) and solved affirmatively by [Lev, Nagy, and Pach](https://mathoverflow.net/a/380683/61536).


**Remarks 2.** For some non-commutative groups like generalized dihedral groups the answer to Problem is also affirmative, see my [partial answer](https://mathoverflow.net/a/396764/61536) below. This partial answer suggests the following



> 
> **Problem 2.** Let $G$ be a group containing an Abelian subgroup of index 2. Is every finite decomposable set in $G$ product-one?
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> 
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Product-one sets in non-commutative groups
GAP shows that the groups SmallGroup(27,3), SmallGroup(27,4), SmallGroup(36,11), SmallGroup(39,1) SmallGroup(48,3) do contain many 5-element decomposable sets, which are not product-one. So, the lower bound 5 for the smallest cardinality of a counterexample, obtained by @YCor in his answer, is the best possible.


Below I write down 5-element decomposable non-product-one sets found by GAP in the groups


SmallGroup(27,3): [ f1, f2, f1 \* f2, f1^2 \* f2, f1 \* f2^2 ]


SmallGroup(27,4): [ f1, f2, f1 \* f2 \* f3, f1^2 \* f2 \* f3, f2^2 \* f3^2 ]


SmallGroup(36,11): [ f1, f2 \* f3, f1^2 \* f3, f1 \* f2^2 \* f3, f1^2 \* f2^2 \* f4 ]


SmallGroup(39,1): [ f1, f2, f1 \* f2, f1^2 \* f2, f2^4 ]


SmallGroup(48,3): [ f1, f2, f1 \* f2, f2 \* f3, f1^2 \* f2 ]


These 5 groups are the only groups of order $\le 50$ that contain decomposable non-product-one sets.


