Power of Product in Abelian Group/Additive Notation
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Theorem
Let $\struct {G, +}$ be an abelian group.
Then:
- $\forall x, y \in G: \forall k \in \Z: k \cdot \paren {x + y} = \paren {k \cdot x} + \paren {k \cdot y}$
Proof
By definition of abelian group, $x$ and $y$ commute.
That is:
- $x + y = y + x$
The result follows from Power of Product of Commutative Elements in Group.
$\blacksquare$
Sources
- 1970: B. Hartley and T.O. Hawkes: Rings, Modules and Linear Algebra ... (previous) ... (next): $\S 2.1$: Subrings: Notation $2$