Index of Subgroup equals Index of Conjugate

Theorem

Let $G$ be a group.

Let $H$ be a subgroup of $G$.

Then:

$\index G H = \index G {a H a^{-1} }$

where $\index G H$ denotes the index of $H$ in $G$.

Proof

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Sources

• 1965: J.A. Green: Sets and Groups ... (previous) ... (next): Chapter $6$: Cosets: Exercise $9$