How to find the group associated with a given symmetry

Question

I have been doing some research as a theoretical physicist and I came across a physical system where any even number of quantum deformations of space leave the system unchanged.Hence two deformations is physically the same as say ten maybe and any other even number so that changing it doesn't change the physical situation.Is there any lie group that can be used to encode this symmetry? Thanks in advance

Answer 1

From a group-theory perspective it seems to me that you are asking about an abstract group $G$ having the property that the product of every even length sequence of nontrivial elements of $G$ is equal to the identity element of $G$. Any such group $G$ is isomorphic to the order 2 cyclic group.

Ordinarily one doesn't think of $G$ as a Lie group, although one could certainly represent $G$ inside some Lie groups. Many Lie groups contain elements of order $2$. For example, in the Lie group $GL(n,\mathbb C)$ you can take $M$ to be any diagonalizable matrix all of whose eigenvalues are either $+1$ or $-1$, with at least one $-1$ eigenvalue, in which case $M^2=I$, the identity matrix, but $M \ne I$. Letting $G = \{M,I\}$ you get a subgroup $G$ of $GL(n,\mathbb C)$ that is isomorphic to the order 2 cyclic group.