Answer:

1.

(I).Items are due to packing of atom/molecules in a crystal.

(a). In three-dimensional space, the equivalent repeat unit is a parallelepiped known as the unit cell.
(b). The crystal lattice itself is actually an infinite set of equally-spaced point.
(d). The origin in crystal occurs inside crystal.

(II). Items are due to chosen by the crystllographyer.

(c). In the reciprocal space of crystal, the origin is an arbitrary point chosen by crystallographyer.

(III). Items are due to chosen by neither (I) or (II).

(e). The Space-Group is a set of elements for which a combination law is defined.

2.

Laue Classes

In agreement with Neumann's principle, physical experiments do not normally reveal the true symmetry of the crystal: some of them, for example diffraction, show the symmetry one would obtain by adding an inversion centre to the symmetry elements actually present. In particular this happens when the measured quantities do not depend on the atomic positions, but rather on the interatomic vectors, which indeed form a centrosymmetric set. Point groups differing only by the presence of an inversion centre will not be differentiated by these experiments.

In a particular experiment the unit-cell geometry may be such that a ¹ b ¹ c and   a = b = g = 90° within experimental error. Typically the esd' on the angles will be relatively large and so to test for orthorhombic symmetry it is necessary to check, say, that I = (hkl) and I = (h-k-l) to demonstrate that twofold symmetric with respect to a and c, respectively. This is often referred to as a Laue check on the symmetry of the crystal.

3.

Those point groups which possess no improper rotations are called enantiomorphic. Enantiomorphic molecules have right-handed and left-handed forms. Normally, the condition for enantiomorphic well shows rotary axes and lack of molecular rotary-reflection axes, which means it cannot be superimposed on to its mirror image in any of its possible conformations. So crystallographyer can not find lattice translations and mirror/glide plans in the crystal structure of an enantiomorphic molecule.

4.

I.

An organometallic cobalt compound crystal which accords space group P21/c has central metal atom, Co with co-ordination 0.271, 0.250,0..458.
In the 3-dimention space of this crystal, the view down b has following symmetry operations:

(1). A lattice translation is parallel to a of (1,0,0). The corresponding symmetry operator is 1+x, y, z (i.e. co-ordination 1.271, 0.250, 0.458).
(2). A lattice translation is parallel to c of (0,0,1). The corresponding symmetry operator is x, y, 1+z (i.e. co-ordination 0.271, 0.250, 1.458).
(3). Two lattice translations: one is parallel to a of (1,0,0) and the other parallel to c of (0,0,1). The corresponding symmetry operator is 1+x, y, 1+z (i.e. co-ordination 1.271, 0.250, 1.458).
(4). A two-one screw axis is parallel to b at (½, y, ¼). The corresponding symmetry operator is 1-x, ½+y, ½-z (i.e. co-ordination 0.729, 0.750, 0.042).
(5). A two-one screw axis is parallel to b at (0, y, ¼). The corresponding symmetry operator is -x, ½+y, ½-z (i.e. co-ordination -0.271, 0.75, 0.042).
(6). There is a centre of inversion at the point (½, 0, ½). The corresponding symmetry operator is 1-x, -y, 1-z (i.e. co-ordination 0.729, -0.250, 0.542).
(7). There is a centre of inversion at the point (0, 0, ½). The corresponding symmetry operator is -x, -y, 1-z (i.e. co-ordination -0.271, -0.250, 0.542).
(8). There is a centre of inversion at the point (1/2, 0, 0). The corresponding symmetry operator is 1-x, -y, -z (i.e. co-ordination 0..729, -0.250, -0.458).
(9). There is a point of inversion at the origin (0, 0, 0). The corresponding symmetry operator is -x, -y, -z (i.e. co-ordination -0.271, -0.250, -0.458).
(10). There is a c-glide plane at (x, ¼, z). The corresponding symmetry operator is x, ½-y, ½+z (i.e. co-ordination 0.271, 0.250, 0.958).
(11). There is a c-glide plane at (x, ¼, z) plus a lattice translationparallel to a of (1, 0, 0). The corresponding symmetry operator is 1+x, ½-y, ½+z (i.e. co-ordination 1.271, 0.25, 0.958).

Views down c:

(1). There is a lattice translation parallel to b of (0, 1, 0). The corresponding symmetry operator is x, 1+y, z (i.e. co-ordination 0.271, 1.25, 0.458).
(2). There is a lattice translation parallel to a of (1, 0, 0). The corresponding symmetry operator is 1+x, y, z (i.e. co-ordination 1.271, 0.25, 0.458).
(3). There are two lattice translations: one is parallel to a of (1, 0, 0) and the other is parallel to b of (0, 1, 0). The corresponding symmetry operator is 1+x, 1+y, z (i.e. co-ordination 1.271, 1.25, 1.458).
(4). There is a two-one screw axis parallel to b at (½, y, ¼). The corresponding symmetry operator is 1-x, ½+y, ½-z (i.e. co-ordination 0.729, 0.750, 0.042).
(5). There is a two-one screw axis parallel to b at (0, y, ¼). The corresponding symmetry operator is -x, ½+y, ½-z (i.e. co-ordination -0.271, 0.750, 0.042).
(6). There is a centre of inversion at the point (½, ½, 0). The corresponding symmetry operator is 1-x, 1-y, -z (i.e. co-ordination 0.729, 0.750, -0.458).
(7). There is a centre of inversion at the point (½, 0, 0). The corresponding symmetry operator is 1-x, -y, -z (i.e. co-ordination 0..271, -0.25, -0.458).
(8). There is a centre of inversion at the point (0, ½, 0). The corresponding symmetry operator is -x, 1-y, -z (i.e. co-ordination -0.729, 0.25, -0.458).
(9). There is a centre of inversion at the origin (0, 0, 0). The corresponding symmetry operator is -x, -y, -z (i.e. co-ordination -0.271, -0.25, -0.458).
(10). There is a c-glide plane at (x, ¼, z). The corresponding symmetry operator is x, ½-y, ½+z (i.e. co-ordination 0.271, 0.25, 0.958).
(11). There is a c-glide plane at (x, ¼, z) plus a lattice translation parallel to a of (1, 0, 0). The corresponding symmetry operator is 1+x, ½-y, ½+z (i.e. co-ordination 1.271, 0.25, 0.958).

Views down a:

(1). There is a lattice translation parallel to b of (0, 1, 0). The corresponding symmetry operator is x, 1+y, z (i.e. co-ordination 0.271, 1.25, 0.458).
(2). There is a two-one screw axis parallel to b at (0, y, ¼). The corresponding symmetry operator is -x, ½+y, ½-z (i.e. co-ordination -0.271, 0.75, 0.042).
(3). There is a centre of inversion at the point (0, ½, ½). The corresponding symmetry operator is -x, 1-y, 1-z (i.e. co-ordination -0.271, 0.250, 0.042).
(4). There is a centre of inversion at the point (0, 0, ½). The corresponding symmetry operator is -x, -y, 1-z (i.e. co-ordination -0.271, -0.25, 0.542).
(5). There is a c-glide plane at (x, ¼, z). The corresponding symmetry operator is x, ½-y, ½+z (i.e. co-ordination 0.271, 0.250, 0.958).
Therefore, co-ordinations Co atoms within the unit cell present above possible co-ordinations.

II.

In the crystal structures of organic and organometallic compounds, it is very common for the molecules to occupy general positions within the unit cell unless the molecule possesses a point-group symmetry witch can coexist with some of the point-group symmetry elements of the space group. When atoms are positioned on point-group symmetry elements of a space group, they are said to be on special positions. According to that, in above crystal Co atoms are not sited on point-group symmetry element of a P2₁/c space group.

5.

In 1,2-dichloroethanein conformation only rotation symmetry element is present. Hence, the molecular point symmetry for this particular conformation is -1.

6.

I.
There are three types of symmetry element present in the diagram presented in problem set 4
(1). Two-fold element
(2). Mirror element
(3). a-glide element

II.
The position of symmetry-related objects within the unit cell is specified in following diagram.

III.
For each symmetry-related object specified in (II) above, there are four pairs of handedness (i.e. original object and Z, g and h, B and C, and J and K) due to the mirror symmetry element.

IV.
The crystal system probably belongs to P2ma, because there are three symmetry elements in the space group and it also is primitive space group which has been specifiey.