Is every pure set of permutations a group?

Question

Let $\mathcal{P}$ be the set of permutations over a finite set $\mathcal{S}$, with $|\mathcal{P}|$=$|\mathcal{S}|!$

$(\mathcal{P},\circ)$ is a finite group, where $\circ$ is composition.

A subset $\mathcal{Q}$ of $\mathcal{P}$ is defined as pure when $\forall (a,b,c)\in\mathcal{Q}^3, a\circ b^{-1}\circ c\in \mathcal{Q}$ (update: and $\mathcal{Q}$ is non-empty).

Any (update: non-empty) subset $\mathcal{Q}$ of $\mathcal{P}$ that is closed under $\circ$ is a (finite, sub)group, and is thus pure.

Is every pure set of permutations a group? If not except for trivial $|\mathcal{S}|$, what's a minimal counterexample? (Solved, see update below).

Any idea to determine if the set $\mathcal{Q}$ of permutations resulting from the cryptographic function DES is pure, with application to the security of 3DES, as asked in this question?

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Update: a minimal counterexample occurs when $\mathcal{P}$ is the set of the 2 permutations of 2 elements; and $\mathcal{Q}$ is the set consisting of the single non-identity permutation, which is pure, but not a group.

Marc van Leeuwen states in his answer that a pure subset $\mathcal{Q}$ of $\mathcal{P}$ (assumed finite) is a group iff it contains the identity. Proof in the not-quite-trivial direction: if $\mathcal{Q}$ is pure and contains the identity $e$, then

• by setting $a=c=e$, $\mathcal{Q}$ contains the inverse of each of its element $b$;
• by setting $b=e$, $\mathcal{Q}$ is closed under $\circ$;
• thus $\mathcal{Q}$ obeys all the properties of a group.

Even combined with the established fact that DES is not a group, and that it would be possible to test/prove that DES does not contains identity, that does not seem to prove that DES is not pure.

Answer 1

The empty set and all one-element subsets of $\mathcal P$ are pure.

Answer 2

$$Q:=\{(12)\}\subset S_3\;:(12)(12)^{-1}(12)=(12)\in Q$$

yet $\,Q\,$ is not a subgroup...

Answer 3

Under the assumption that $\mathcal P$ is finite, a pure subset is a subgroup if and only if it contains the identity permutation$~e$ (and you only need instances with $c=e$ in the description of pure subsets to show this).