$\{5,15,25,35\}$ is a group under multiplication mod $40$

Solution 1:

I think all you need has been pointed here. You don't need to write down the associated Cayley table for the presented set and its operation, but for this one it leads you to get the answer graphically:

enter image description here

We can see that the operation is a binary one. Can you find the identity element? What about the inverses ones?

Solution 2:

If you'd have {1,3,5,7} as a group under multiplication modulo 8 the canonic identity would be the 1. Since all element as well as the modulo factor are multiplied by 5 (twice for the product) the identity "shifts" to 25. The "relatively prime" statement works for the simple {1,3,5,7} case since they are relatively prime to 8 as well, but not for {5, 15, 25, 35} since they are not relatively prime to 40. I don't know enough of the structure of unitary groups to see a relationship to your problem.

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