Is 著名 a valid simplified word or is it 著着?

I learned 著名 (Zhùmíng) from HelloChinese using the simplified form. However, Google Translate and many other resources all say that 著 is a traditional character which is simplified into 着. To make it more confusing, Google Translate and others seem to think that 著名 is the same in both simplified and traditional.

Is 著名 a legitimate simplified word?

If so, what is the deal with this 著 character. When does it remain as 著 and when is it simplified to 着?

submitted by /u/chinawcswing
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