Powers of Group Elements/Product of Indices
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Theorem
Let $\struct {G, \circ}$ be a group whose identity is $e$.
Let $g \in G$.
Then:
- $\forall m, n \in \Z: \paren {g^m}^n = g^{m n} = \paren {g^n}^m$
Additive Notation
This can also be written in additive notation as:
- $\forall m, n \in \Z: n \cdot \paren {m \cdot g} = \paren {m \times n} \cdot g = m \cdot \paren {n \cdot g}$
Proof
All elements of a group are invertible, so we can directly use the result from Index Laws for Monoids: Product of Indices:
- $\forall m, n \in \Z: g^{m n} = \paren {g^m}^n = \paren {g^n}^m$
$\blacksquare$
Also see
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: $(1.6)$
- 1965: J.A. Green: Sets and Groups ... (previous) ... (next): $\S 4.6$. Elementary theorems on groups: Example $87 \ \text{(vii)}$
- 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 35.10 \ \text {(ii)}$: 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 $3.8 \ (2)$