(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Take G to be the cyclic group with 12 elements. Find an element g in G such that the equation x^2 = g has no solution.

2. Relevant equations

Notation: Z = set of integers.

A group is said to be commutative or Abelian if the operation * satisfies the commutative law, that is, if for all g and h in G we have g*h=h*g.

Some quotes from my textbook with relevant information:

"Let G be a group and let g be an element of G. The set <g> = {g^n: n in Z} of all distinct powers of g is a subgroup, known as the subgroup generated by g. It has n elements if g has order n and it is infinite if g has infinite order.

A group of the above type, that is, of the form <g> for some element g in it, is said to be cyclic, generated by g.

Remark: It follows from the above theorem that a cyclic group is Abelian."

3. The attempt at a solution

I'm really confused with this entire chapter. It says G is cyclic with 12 elements, so does it look like this?

G = {g^0, g^1, ... , g^10, g^11},

such that g^12 = g^0, g^13 = g^1, etc. (?)

But what is the question asking for? And how does it follow that a cyclic group is Abelian?

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# Homework Help: Help! Group Theory Question(s)

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