Knowlpedia

Conjugation preserves order

Conjugate elements in a group have the same order.

Proposition (Conjugation preserves order). Let GG be a . If x,y∈Gx,y\in G are , so that y=gxg−1y=gxg^{-1} for some g∈Gg\in G, then

ord⁡(y)=ord⁡(x),\operatorname{ord}(y)=\operatorname{ord}(x),

where an element has order ∞\infty if no positive power of it is the identity.

Why this holds

For every integer nn, one has yn=gxng−1y^n=gx^ng^{-1}. Hence yn=ey^n=e exactly when xn=ex^n=e, so the two elements have the same finite order or both have infinite order.