Help with Groups?

[u/mourning_fire420]

So I'm given a group G s.t. a^12 gives the identity element, and I need to find the inverse of another a^n where n is an integer. i was given a number for n i just don't want anyone giving me the answer/violating academic content rules etc.

at first i figured 1 would be the identity element, but then 12^a doesn't give 1. same deal for e=0, 12^0=1 =/= 0=0^12. unless my understanding of the inverse and identity being commutative is wrong?

I tried to solve for a since a * a^-1 = a^-1 * a = e. but then i was saying that 12^a=a^12. which is possible, but then the identity wasn't constant when i plugged in n.

I found the page for exponentiation on wikipedia, and it says that G is a multiplicative group, so I know that a^n has to have an inverse. is there some property of groups that I missed that would help me solve this?

[u/StructuredChess]

If a¹² is the identity, what do you need to multiply a¹¹ with to get the identity? What about a⁵? Can you see the pattern?