# Proof of Pauli group preservation by Clifford group conjugation?

A well know result is that Clifford group preserve the Pauli group under conjugation or, in other words:

• $C(P_{1} \otimes P_{2})C^{\dagger} = P_{3} \otimes P_{4}$, with $C \in$ Clifford group and $P_{n} \in$ Pauli group.

How we can prove this?

Thank's...

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Can you say some more about what these groups are? – Ryan Thorngren Sep 12 '12 at 4:22
This actually seems like more of a pure math question to me, and that suggests it may get a better response on Mathematics... I can migrate it if people think that would be appropriate. – David Z Sep 12 '12 at 5:00
Also, it seems like a relatively trivial problem of multiplying finitely many matrices, but please define these to get a proper answer. You should know that "preserving a group" usually means preserving the algebraic relations, and then all conjugations do that. In this case, I assume you mean multiplying and product of $\sigma_x,\sigma_y,\sigma_z$ by some finite set of matrices and their inverse keeps you in the set of these matrices and their products – Ron Maimon Sep 12 '12 at 6:33