Product of Conjugates equals Conjugate of Products
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Theorem
Let $\struct {G, \circ}$ be a group.
Then:
- $\forall a, x, y \in G: \paren {a \circ x \circ a^{-1} } \circ \paren {a \circ y \circ a^{-1} } = a \circ \paren {x \circ y} \circ a^{-1}$
That is, the product of conjugates is equal to the conjugate of the product.
Proof
Follows directly from the group axioms.
$\blacksquare$
Sources
- 1965: J.A. Green: Sets and Groups ... (previous) ... (next): $\S 6.6$. Normal subgroups: Example $124 \ \text{(i)}$