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Proof

(related to Corollary: Rules for Exponentiation in a Group)

Ad $(1)$

• The rule follows immediately from the definition of exponentiation in a group, the associativity of the operation "$\ast$" and the associativity of adding integers.

Ad $(2)$ and $(3)$

• These rules require in addition the commutativity of the operation "$\ast$" and follow from the commutativity of adding integers.

Ad $(4)$

• The rule is only a special notation for the rule $(3)$, since $$\frac{x^n}{y^n}=x^n\ast y^{-n}=x^n\ast (y^{-1})^n=\left(x\ast y^{-1}\right)^n.$$
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References

Bibliography

1. Forster Otto: "Analysis 1, Differential- und Integralrechnung einer Veränderlichen", Vieweg Studium, 1983