# How can I verify that the Pauli group is a group? And is it abelian? [duplicate]

So how can I verify that the Pauli Group is a Group? Then furthermore, Abelian? And then to sum it up, the order of the group. Trying to do some research into the group but I can't find much about it.

• What makes you think it’s abelian? Nov 24, 2020 at 4:35
• en.wikipedia.org/wiki/Pauli_group Nov 24, 2020 at 5:57
• Hasan how is that useful in anyway to what was asked? Anyone can copy and paste a link from a wiki that literally explains nothing. Nov 24, 2020 at 5:59
• And Mark that's what I was asking, is it Abelian haha Nov 24, 2020 at 6:00
• Using the link @HasanIqbal gave you, you could count the number of elements in the set and multiply a few of them together to check if it were abelian or not. The link pretty much answers your question. Nov 24, 2020 at 8:49

For the three Pauli matrices, {$${\sigma_1,\sigma_2}=0$$}, so certainly this can not form an abelian group. The Pauli group is an isomorphism with $$D_4$$.
The elements of the Pauli group are {$${I, \sigma_x, \sigma_y, i\sigma_z, -I, -\sigma_x, -\sigma_y, -i\sigma_z}$$}, so the order of this group is 8. In variant subgroups are {$${I, -I}$$}, {$${I, -I, \sigma_{x/y}, -\sigma_{x/y}}$$} and {$${I, -I, i\sigma_z, -i\sigma_z}$$}. $$I$$ and $$-I$$ is the only emelemt in its class. {$$\pm\sigma_{x/y}$$} and {$$\pm i\sigma_z$$} forms a class( totally 5 classes). Order 1 element: $$I$$. Order 2 elements: $$\sigma_x, \sigma_2, -I, -\sigma_x, -\sigma_y$$. And Order 3 elements: $$\pm i\sigma_z$$.
You can also see Wikipedia and check it is identical to the group table of $$D_4$$(in fact the tables are different because the ordering of elements is different).