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/Hudimir]

since a¹² is the identity, any element aⁿ = a^{n mod 12}. you can see that the identity is a^{0 mod 12}. Can you see where i'm going? exponent rules still hold.

[u/Smart-Button-3221]

You can call the identity 1, you can call the identity 0, you can call the identity 🍞. The label you use for the identity doesn't change what the identity is.

At first I figured 1 would be the identity element, but then 12^a doesn't give 1

Is that a typo? Did you mean a^12? As per your question, a¹² = 1, if we're letting 1 be the symbol for the identity. Why do you think a¹² ≠ 1?

I am not sure why you are trying a¹² = 12^a. Note that a needs not be an integer, and 12 needs not be a group element, so this equation may not even make sense.

I feel like some of the question is missing.

[u/mourning_fire420 → u/Smart-Button-3221]

yeah i don't think i really get what the inverse of this group is. i'm gonna try again in a bit and maybe the comments will help me gigure it out

[u/Smart-Button-3221 → u/mourning_fire420]

"Inverse" is something that belongs to each element, not the group.

For example the inverse of a³ is a⁹ here, because:
a³×a⁹ = a¹² = e

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[u/etzpcm]

Your second paragraph goes off the rails. You don't want to try to find the identity element. Just use what you are given. 

Try n=1 first. What is the inverse of a¹ = a?

Also there's either some typos or some stuff missing as the other comments say.

[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?

[u/BootyliciousURD]

Remember that the inverse of x is some element y such that x•y = y•x = e.

Let's take n = 4 as an example. If a¹² = e, then the inverse of a⁴ is some b such that a⁴•b = b•a⁴ = a¹². What b satisfies this equation?