# Weak subgroups

Here is a teatime puzzle.

Suppose ${G}$ is a finite group and ${S \subseteq G}$ is a subset containing the identity such that ${xy \in S}$ whenever ${x \in S}$, ${y \in S}$, and ${x \neq y}$. Show that these are the only possibilities:

1. ${S}$ is a subgroup,
2. ${S = \{1, x\}}$ for some ${x \in G}$,
3. ${S = \{1, x, x^{-1}\}}$ for some ${x \in G}$,
4. ${S = Q \setminus \{-1\}}$ for some quaternion subgroup ${Q \leq G}$.

What other possibilities are there if we do not assume ${1 \in S}$?