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I am new at group theory, and I came across a question I would like help with

Suppose we have a set $S$ with the only elements $p,q,r$. Let $a$ and $b$ be two elements of $S$. Consider the following properties of $S$:

1) $aa=a$

2) $ab=ba$

3) $(ab)c=a(bc)$

4)$pa=a$ for every element $a$

Prove that there exists some element in $b \in S$ such that $bp=b, bq=b, br=b$.

Thank you! I am new in Group Theory so i was just looking for some help

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    I assume that you were also told that the "product" of any two elements of $S$ is in $S$. Note then that we know almost all the "multiplication table" for $S$, the only thing missing is what is $qr$. Could $qr$ be $p$? Could it be $q$? $r$? Hint: Use 4).2012-01-28
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    @André: I don’t think that you can determine which of the non-identity elements $qr$ is.2012-01-28
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    @Brian M. Scott: One cannot, but there is symmetry, so we don't care.2012-01-28
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    @André: Agreed. It just seemed to me that a natural interpretation of the hint for someone having trouble with the original problem would be that you *can* determine that product.2012-01-28
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    I changed the tags from [group-theory] and [finite-groups] to [semigroups], because no group can satisfy the given conditions.2012-01-28

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