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- M is a
*t*-subgroup (*translationengleich*subgroup) in G: [*i*] is the_{t}*t*-index of M in G - H is a
*k*-subgroup (*klassengleich*subgroup) in M: [*i*] is the_{k}*k*-index of H in M

As a consequence of Hermann Theorem one can distinguish between three types of subgroups:

(also called translation equivalent subgroups):*t*-subgroups

If H is a*t*-subgroup of G means that T(H) = T(G),*i.e.*the translation groups of G and H are the same.

The subgroup H loses rotation type operations with repect to G and therefore the point group P(H) < P(G)

M coincides to H.(also called point group equivalent subgroups):*k*-subgroups

If H is a*k*-subgroup of G means that P(H) = P(G),*i.e.*the point groups P(G) and P(H) are the same.

The subgroup H loses translation type operations with repect to G and therefore the translation group T(H) < T(G)

M coincides to G.**general type subgroups**: T(H) < T(G) and P(H) < P(G) and so, H < M < G

In the formula, [

In terms of the corresponding indices:

(called also translation equivalent subgroups):*t*-subgroups*i*= 1, then_{k}*i*=*i*_{t}

(called also point group equivalent subgroups):*k*-subgroups*i*= 1, then_{t}*i*=*i*_{k}

**general type subgroups**:*i*≠ 1 and_{k}*i*≠ 1_{t}

In the given formula [

where N

Neglecting lattice deformation, the index [

Consider the phase transition between tetragonal *P*4*mm* (No. 99) and cubic *Pm*-3*m* (No. 221) structures of BaTiO_{3}. The number of atoms *per conventional unit cell* are equal to N_{c}(G) = N_{c}(H) = 5. The two phases are group-subgroup related, and a valid index for this transformation according to the formula is equal to the ratio between the orders of the point groups |P(G)| / |P(H)| (the order of a point group is just the number of the elements of the point group). The ratio between the orders of the point groups can be obtained from the program POINT. The valid index for this transformation is 6.

Consider the group-subgroup related phase transition between orthorhombic *Pnma* (No. 62) and *Pna2*_{1} (No. 33) phases of K_{2}SeO_{4}. The number of atoms *per conventional unit cell* for the given structures are N_{c}(G) = 28 and N_{c}(H) = 84. The orders of the point groups are |P(G)| = 8 and |P(H)| = 4. A simple application of the formula gives an index 6 for this transformation.

- Lattice parameters of K
_{2}SeO_{4}- Pnma: 7.605 10.383 5.949 90 90 90 and V(G) = 469.75 A^{3} - Lattice parameters of K
_{2}SeO_{4}- Pna2_{1}: 22.716 10.339 5.967 90 90 90 and V(H) = 1401.41 A^{3}

Consider the group-subgroup related phase transition between monoclinic *C*2/*c* (No. 15) and *P*2_{1}/*c* (No. 14) phases of CaTiSiO_{5}. The number of atoms *per conventional unit cell* for the given structures are N_{c}(G) = 32 and N_{c}(H) = 32. A simple application of the formula gives an index 2 for this transformation.

**Orientational (twin) domain states**: As a consequence of a lost of rotational elements**Antiphase domain states**: As a consequence of a lost of translational elements

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