Powers of Group Elements/Sum of Indices

Theorem

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

Let $g \in G$.

Then:

$\forall m, n \in \Z: g^m \circ g^n = g^{m + n}$

This can also be written in additive notation as:

$\forall m, n \in \Z: \paren {m \cdot g} + \paren {n \cdot g} = \paren {m + n} \cdot g$

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

$\forall m, n \in \Z: g^m \circ g^n = g^{m + n}$

$\blacksquare$

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: $(1.5)$
1965: J.A. Green: Sets and Groups ... (previous) ... (next): $\S 4.6$. Elementary theorems on groups: Example $87 \ \text{(vi)}$
1970: B. Hartley and T.O. Hawkes: Rings, Modules and Linear Algebra ... (previous) ... (next): $\S 2.1$: Subrings: Notation $2$
1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 35.10 \ \text {(i)}$: 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 $3.8 \ (1)$