# Lie algebra of a finite group

I'm trying to find the normalizer of the Pauli group $G_n$ (as a subgroup of $SU(2^n)$) utilizing Lie algebras, as is done in a reference to find the normalizer of the Heisenberg group $HW(n)$. There, they see its Lie algebra $\mathfrak{hw}(n)$ (the Heisenberg algebra) to figure out a (Lie) algebra containing it as an ideal, that is, the normalizer $N[\mathfrak{hw}(n)]$. The result is the semidirect sum of the Heisenberg algebra and the real symplectic algebra $\mathfrak{sp}(2n,\mathbb{R})$. Then, the corresponding Lie group (called the Clifford group $C_n$ in quantum information community) amounts to the sought-after normalizer (of the Heisenberg group).

In applying the same line of techniques to the Pauli group, I bump into the fact that the Pauli group is not a continuous group in the first place (though, it might be a discrete subgroup of $SU(2^n)$).

Q.) Do we have a Lie-algebra-like things for the Pauli group? More generally, do we have a linear space or algebra (such as Lie algebras) to understand a finite group that enables similar techniques mentioned to find its normalizer? As a side-question, what is ordinary method to find the normalizer of a group in general?

• Please try to write your question as self-contained as possible. E.g. for a question about a normalizer to make sense, you need to specify a larger group $H$ containing $G_n$; then ask for $N_H(G_n)$. What should that group be in this case? – Max Jan 11 '16 at 9:06
• Dear Max, I tried to improve my question. While the larger group for the @Max Pauli group regarding its normalizer is $SU(2^n)$ (since I'm a quantum information scientist), I couldn't find any name for that of the Heisenberg group, which is the whole unitary group acting on the infinite-dimensional separable Hilbert space (the Schrödinger space). Thank you for your comment. – eneron Jan 15 '16 at 5:02