Understanding groups with binary operation

Question

With just being done covering groups, there was an exercise in my class notes that we didn't get the chance to cover:

My instructor gave us sample sequences to test to see if they can occur or not, so I looked at one:

So I was trying to wrap my head around the sequence of a increasing to the sixth power and viewed them as orders, from order 1 to 6. Is there something I'm overlooking? I am kind of lost as to how someone would go about getting a sequence based on the orders of a and a binary operation.

Thanks for reading and helping!

Answer 1

I'm assuming you mean that in chronological order, these sequences are the same. That is to say:

$a=a$,

$a^2=b$,

$a^3=c$,

$a^4=e$,

$a^5=a$,

$a^6=d$.

From this, we get a series of equalities: $d=a^6=a^4a^2=ea^2=a^2=b$. Given that, what can you conclude about the sequence a, b, c, e, a, d?

Answer 2

I'm not quite sure what you are looking for in an answer, but I can give you some guidance and information that may be useful.

One thing to consider is that the order of $a$ must divide the order of the group by Lagrange's theorem. In your case, the order of the group is $5$, so the order of $a$ must be $1$ or $5$. The order of $a$ is not $1$ because $a\not= e$, so the order of $a$ is $5$. Therefore the example sequence your instructor gave you is impossible since it would imply that $a^4 = e$. I could say some more, but you can probably figure the rest out on your own. Hope this helps.