Proof
(related to Proposition: Simple Calculations Rules in a Group)
Ad 1)
- Because of the existence of inverse elements it is \(e=e\ast e^{-1}\).
- On the other hand, because $e$ is the unique neutral element, we have \(e^{-1}=e\ast e^{-1}\).
- Comparing both equations leads to \(e^{-1}=e\).
Ad 2)
- The element \(x^{-1}\) has the inverse elements \((x^{-1})^{-1}\) and \(x\), respectively.
- Both must be identical, since inverse element are unique.
Ad 3)
- \(a^{-1}\ast b\) is the solution of \(a\ast x=b\), since \(a\ast (a^{-1}\ast b)=(a\ast a^{-1})\ast b=e\ast b=b\) for all \(a,b\in G\).
- Moreover, it is the only solution. For if an element $y\in G$ solves \(a\ast y=b\), it follows
\[\begin{array}{rcl}
a\ast y&=&b\\
a^{-1}\ast (a\ast y)&=&a^{-1}\ast b\\
(a^{-1}\ast a)\ast y&=&a^{-1}\ast b\\
e\ast y&=&a^{-1}\ast b\\
y&=&a^{-1}\ast b
\end{array}
\]
Ad 4)
Exercise, in analogy to 3)
Ad 5)
- If $a\ast x=a\ast y$ then for all $a\in G$:
\[\begin{array}{rcl}
a\ast x&=&a\ast y\\
a^{-1}\ast (a\ast x)&=&a^{-1}\ast (a\ast y)\\
(a^{-1}\ast a)\ast x&=&(a^{-1}\ast a)\ast x\\
e\ast x&=&e\ast y\\
x&=&y
\end{array}
\]
- A similar cancellation property can be concluded for the equation $x\ast a=y\ast a$, from which it follows $x=y$ for all $a\in G.$
Ad 6)
The element \((x\ast y)\) has the inverse element \((x\ast y)^{-1}\). It follows for all \(x,y\in G\):
\[\begin{array}{rcl}
(x\ast y)\ast(x\ast y)^{-1}&=&e\\
y^{-1}\ast x^{-1}\ast (x\ast y)\ast(x\ast y)^{-1}&=&y^{-1}\ast x^{-1}\ast e\\
y^{-1}\ast (x^{-1}\ast x)\ast y\ast(x\ast y)^{-1}&=&y^{-1}\ast x^{-1}\\
y^{-1}\ast e\ast y\ast(x\ast y)^{-1}&=&y^{-1}\ast x^{-1}\\
y^{-1}\ast y\ast(x\ast y)^{-1}&=&y^{-1}\ast x^{-1}\\
(y^{-1}\ast y)\ast(x\ast y)^{-1}&=&y^{-1}\ast x^{-1}\\
e\ast(x\ast y)^{-1}&=&y^{-1}\ast x^{-1}\\
(x\ast y)^{-1}&=&y^{-1}\ast x^{-1}\\
\end{array}
\]
Ad 7)
- "$\ast$" is associative by definition of a group (a group is a monoid, which is a semigroup, which is associative)
- Therefore, we can apply the general associative law we have already proven for any associative algebraic structure.
Ad 8)
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