The chain is wider than cog, it's safe to ride, and doesn't fall over, it's staying properly, chain is 1/2 inch pitch x 1/8, and my cog is 1/2 pitch x 3/32.
Is it okay that cog is thinner than the chain ?
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Yes, that's normal. Todays sprockets are made to be usable with thin chains, but there's also an older single speed standard out there that had wider chains. You can safely ride a modern, thin sprocket with a classic wide chain.
The important thing is, that the length of the chains links agrees with the teeth of the sprocket, but its width is relatively irrelevant.