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To complete the data we need for classifying the groups, we need one more concept to add to the lattice *L*, the point group *P*, and the action of *P* on *L*. This is to allow for the fact that some reflections in *P* act as glides rather than as reflections in *G*.

**Definition**

Let *A* ∈ *P* be a reflection with *T*_{v} ∘ *A* ∈ *G*. Then the vector ** a** =

**Properties**

*A shift vector lies in the lattice L*.**Proof**

(*T*_{v}∘*A*)^{2}() =*x**T*_{v}∘*A*(+*v**A*()) =*x**T*_{v}(*A*() +*v*) since*x**A*^{2}=*I*and this is*T*_{v+A(v)}() and so*x*+*v**A*() is in the lattice.*v*

*A shift vector of A is left fixed by A*.**Proof**

*A*() =*a**A*(+*v**A*()) =*v**A*() +*v**A*^{2}() =*v**A*() +*v*since*v**A*^{2}=*I*and this isagain.*a*

One can in fact define a shift vector for *any* element of *P*.

For example, if *R* is rotation by 2*π*/*n* and *T*_{v} ∘ *R* ∈ *G* then take ** a** =

**Examples**

- Consider the group of symmetries of this pattern (
**pm**).

The lattice is*rectangular*.

The point group is*D*_{1}which acts on the lattice as vertical reflection*R*.

In this case since*R*is a symmetry,**0**is a shift vector.

To get another shift vector, take*v*as shown so that*T*_{v}is in the group (andis in the lattice) and take*v*=*a*+*v**R*() as shown.*v*

Note thatis on the mirror and so is left fixed by*a**R*.

In this case, ifis a vertical basis element of the lattice, any shift vector of*w**R*is of the form 2*n*where*w**n*∈**Z**.

- The group of symmetries of this pattern (
**pg**).

The lattice is again*rectangular*.

The point group is*D*_{1}which acts on the lattice as vertical reflection*R*.

In this case*R*is*not*a symmetry of the pattern.

To get another shift vector, take*v*as shown so that*T*_{v}is in the group (andis in the lattice) and take*v*=*a*+*v**R*() as shown.*v*

As beforeis on the mirror and so is left fixed by*a**R*.

Then, ifis a vertical basis element of the lattice, any shift vector of*w**R*is of the form (2*n*+1)where*w**n*∈**Z**.

- The group of symmetries of this pattern (
**cm**).

This time lattice is*rhomboid*.

The point group is still*D*_{1}and still acts on the lattice as vertical reflection*R*.

The vector**0**is a shift vector of*R*.

As before, take*v*as shown so that*T*_{v}is in the group (andis in the lattice) and take*v*=*a*+*v**R*() as shown.*v*

As beforeis on the mirror and so is left fixed by*a**R*.

Then, ifis a vertical basis element of the lattice, then shift vector of*w**R*is of the form*n*where*w**n*∈**Z**. In this case*all*the lattice points on the mirror are shift vectors.

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