Power of Coset Product is Coset of Power
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Theorem
Let $\struct {G, \circ}$ be a group.
Let $N$ be a normal subgroup of $G$.
Let $a \in G$.
Then:
- $\forall n \in \Z: \paren {a \circ N}^n = \paren {a^n} \circ N$
Proof
From Quotient Group is Group, the operation:
- $\forall a, b \in G: \paren {a \circ N} \circ \paren {b \circ N} = \paren {a \circ b} \circ N$
is the group operation in the quotient group $\struct {G / N, \circ}$.
The result follows directly by definition of power of group element.
$\blacksquare$
Sources
- 1965: J.A. Green: Sets and Groups ... (previous) ... (next): Chapter $6$: Cosets: Exercise $13$