Generators D2h

M. Kohlhaas

Hello,

in my Book is a formula, namely

sigma^(1) C_2 = sigma^(12)

where sigma^(1) is a reflection about the x-z-plane, C_2 is a pi-rotation about the z-axis and sigma^(12) is a reflection about the midway plane between x-z- and y-z-plane. When i in my Imagination make the steps on the left I come out at a totally different point as where the right-hand side would send me. Obviously I have completely misunderstood the formula and I don't the the sun anymore. Can someone please explain me the meaning of that statement?

Thanks alot

CompuChip

Consider a unit cube and label its corners 1 through 6. Then you can write down how the symmetries act on these corners as a permutation.
Also, note that there is no general consensus on the order of the operations, some people mean "first do [itex]\sigma^1[/itex] then [itex]C_2[/itex]" when they write down [itex]\sigma^1 C_2[/itex], some people read it as "perform [itex]\sigma^1[/itex] after [itex]C_2[/itex]". Maybe this is why you don't get the same answer?

Alternatively, you could write out the operations as matrices working on the vectors of a standard orthonormal basis and work out the left hand side by matrix multiplication.

Gokul43201

Either it's wrong or there's some error in transcribing it. The LHS is equivalent to a reflection about the y-z plane.

Can you include the complete excerpt from the book - this will provide some context and make it possible to debug. Also include the name and edition of the book. The connection between the above operations and the symmetry group in the title (D2h) is also not clear.

M. Kohlhaas

In this case the convention is to first apply [itex]C_2[/itex] and then [itex]\sigma^(1)[/itex].


The book's name is "symmetry - an introduction to group theory" written by Roy McWeeny. Hier is an excerpt; the certain special thing which i asked for is marked in red: