I'm shocked at the 'suggestion' that he deliberately cheated. = The very 'suggestion' that he deliberately cheated is shocking (to me).
They were careful to avoid any 'suggestion' of impropriety. [=avoid doing anything that could be seen as improper]
There was some 'suggestion' of swelling around his ankle. [=his ankle appeared to be slightly swollen]
(..) Good luck, but be careful. - You know you're being shadowed? - It becomes instinct. I observe that you, in one respect, are very fortunate. I'll make one suggestion. Why, I do not know. It cannot possibly profit me. Have you heard about the letters of transit? Yes, something. (..)1942 Casablanca