Problem:Determine whether the binary operation gives a group structure on the given set.

Let be defined on by letting .

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My attempt:

The axioms for a group structure are:

1) Associativity

2) Existence of an identity element (which I will call )

3) Existence of an inverse

For the first axiom - associativity:

Since the entire expression (for each side) is enclosed by absolute value, the inner absolute values serve no purpose and can be removed. Therefore,

and it is shown that is associative.

For axiom 2 - existence of an identity element:

However,

since if is negative , the absolute value of will not equal .

Therefore, the binary operation does not have an identify element and fails axiom two, and so this does not form a group structure.

For axiom three - existence of an inverse:

And therefore an inverse does exist, so axiom three passes.

In conclusion, axiom 1 passes, axiom two fails and axiom three passes. Since the second failed, this is not a group structure.

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Did I do everything correctly? I have the nagging suspicion that I made an error. Any help is greatly appreciated.