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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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    The standard modern single-speed chain is speced as 1/8". 3/32" chains are used on derailleur bikes, except that 0.09" chains are used on bikes with 9 or more rear cogs. The measurement is between the inside surfaces of the narrow link. Note that the cog dimensions will be thinner, to allow clearance. – Daniel R Hicks Jun 20 at 0:33
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    The cog MUST be thinner than the chain, or the chain would get stuck. – Daniel R Hicks Jun 22 at 21:53
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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.

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    Yup - 99.9% of the bicycle chains out there are 1/2 inch pitch. Some very rare 10 and 12mm pitch systems existed in the 80s thanks to Shimano trying to metrify (metricify ?) everything, but they never took off. So all modern bike chain is half-inch. – Criggie Jun 22 at 23:38

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