Problem:Determine whether the binary operation $\displaystyle *$ gives a group structure on the given set.

Let $\displaystyle *$ be defined on $\displaystyle \mathbb{C}$ by letting $\displaystyle a*b=|ab|$.

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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 $\displaystyle e$)

3) Existence of an inverse

For the first axiom - associativity:

$\displaystyle (a*b)*c=a*(b*c)$

$\displaystyle |ab|*c=a*|bc|$

$\displaystyle ||ab|c|=|a|bc||$

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

$\displaystyle |abc|=|abc|$

and it is shown that $\displaystyle *$ is associative.

For axiom 2 - existence of an identity element:

$\displaystyle e*x=x*e=x$

$\displaystyle e*x=|ex|=|x|$

However,

$\displaystyle |x| \neq x$

since if $\displaystyle x$ is negative , the absolute value of $\displaystyle x$ will not equal $\displaystyle x$.

Therefore, the binary operation $\displaystyle *$ 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:

$\displaystyle a*a' = a'*a = e$

$\displaystyle |a\frac{1}{a}|=|\frac{1}{a}a|=1$

$\displaystyle |\frac{a}{a}|=|\frac{a}{a}|=1$

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.