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sammycaps

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Prove the Generalized Associative Law for Groups (i.e. a finite sum of elements can be bracketed in any way).

The proof is outlined in D & F. I just want to know whether or not one part of my proof is correct.

Originally, I considered defining a directed set and working with that, but this seemed like extra unecessary steps. So, I will instead proceed by starting with the expression a

Then we can see that the construction I've described defines all possible ways to bracket the expression, so we are done.

Is this valid, and would we need to show this indeed defines all bracketing possibilities, or is it clear enough?

The proof is outlined in D & F. I just want to know whether or not one part of my proof is correct.

**Show that for any group G under the operation °, and elements a**_{1},...,a_{n}, any bracketing of the expression a_{1}°...°a_{n}can be rewritten as (a_{1}°...°a_{k})°(a_{k+1}°...°a_{n}).Originally, I considered defining a directed set and working with that, but this seemed like extra unecessary steps. So, I will instead proceed by starting with the expression a

_{1}°...°a_{n}and bracketing as follows. First place brackets such that every bracket contains exactly two elements of the group and one °. Then proceed by replacing each bracketed item with a b (so replace a_{i}with b_{i}). Then repeat step one. Continue this process, replacing each new bracketed pair with a new indicator. It must terminate when there are two elements left because there are finite elements and operations. Then we are left with x_{i}° y_{i}. So if we substitute back in all the a_{i}along with the brackets, we will get (a_{1}°...°a_{k})°(a_{k+1}°...°a_{n}), where each half is bracketed in some way.Then we can see that the construction I've described defines all possible ways to bracket the expression, so we are done.

Is this valid, and would we need to show this indeed defines all bracketing possibilities, or is it clear enough?

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