Powers of Group Elements/Negative Index

Theorem

Let $\struct {G, \circ}$ be a group whose identity is $e$.

Let $g \in G$.


Then:

$\forall n \in \Z: \paren {g^n}^{-1} = g^{-n} = \paren {g^{-1} }^n$


Additive Notation

This can also be written in additive notation as:

$\forall n \in \Z: -\paren {n g} = \paren {-n} g = n \paren {-g}$


Proof

All elements of a group are invertible, so we can directly use the result from Index Laws for Monoids: Sum of Indices:

$\forall n \in \Z: \paren {g^n}^{-1} = g^{-n} = \paren {g^{-1} }^n$

$\blacksquare$


Sources

  • 1964: Walter Ledermann: Introduction to the Theory of Finite Groups (5th ed.) ... (previous) ... (next): Chapter $\text {I}$: The Group Concept: $\S 2$: The Axioms of Group Theory: $(1.10)$
  • 1965: J.A. Green: Sets and Groups ... (previous) ... (next): $\S 4.6$. Elementary theorems on groups: Theorem $\text{(v)}$
  • 1971: Allan Clark: Elements of Abstract Algebra ... (previous) ... (next): Chapter $2$: The Definition of Group Structure: $\S 27$
  • 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 35$: Elementary consequences of the group axioms
  • 1996: John F. Humphreys: A Course in Group Theory ... (previous) ... (next): Chapter $3$: Elementary consequences of the definitions: Proposition $3.8 \ (3)$
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