Trans-1,2-difluoroethene ( in the trans

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Group theory and symmetry · Group theory and symmetry

Invariance, Ch Groups: the algebra of symmetry operations

Problem

trans-1,2-difluoroethene (

\mathrm{FHC=CHF}

in the trans, or

E

, configuration) is planar and centrosymmetric. Take the molecular plane as the

xy

-plane and the origin at the midpoint of the

\mathrm{C=C}

bond, which is the molecule's centre of inversion; the

z

-axis is perpendicular to the plane. Its four symmetry operations are represented, in this frame, by the matrices

E=\begin{pmatrix}1&0&0\\0&1&0\\0&0&1\end{pmatrix},\quad C_2=\begin{pmatrix}-1&0&0\\0&-1&0\\0&0&1\end{pmatrix},

\sigma_h=\begin{pmatrix}1&0&0\\0&1&0\\0&0&-1\end{pmatrix},\quad i=\begin{pmatrix}-1&0&0\\0&-1&0\\0&0&-1\end{pmatrix},

where

C_2

is the

180^\circ

rotation about the

z

-axis,

\sigma_h

is reflection in the molecular plane and

i

is inversion through the centre. ((a)) By multiplying the matrices, construct the full

4\times4

multiplication table of

\{E, C_2, \sigma_h, i\}

, writing

gh

for "

h

first, then

g

". ((b)) Using the table, check the four group axioms and state whether the group is abelian. ((c)) Using Lagrange's theorem, list every order a subgroup of this group could have, and give the actual subgroup(s) of each such order. ((d)) List the left cosets of the subgroup

H=\{E,i\}

in the group.

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