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

The Hermann Theorem can distinguish between three types of subgroups:

**t-subgroups**(or translation equivalent 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 is equal to H.**k-subgroups**(or point group equivalent 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 is equal to G.**general type subgroups**: T(H) < T(G) and P(H) < P(G) and so, H < M < G

In the formula, [ i

In terms of the corresponding indices:

**t-subgroups**(or translation equivalent subgroups): i_{k}= 1, then i_{t}= i

**k-subgroups**(or point group equivalent subgroups): i_{t}= 1, then i_{k}= i

**general type subgroups**: i_{k}≠ 1 and i_{t}≠ 1

In the given formula [ i ] is the index of the transformation, |P(G)| and |P(H)| are the orders of the point groups of the corresponding space groups G and H and Z(G), Z(H) are the number of formula units

Consider the phase transition between tetragonal *P*4*mm* (99) and cubic *Pm*-3*m* (221) structures of BaTiO_{3}. The number of formula units *per primitive unit cell* are equal to Z(G) = Z(H) = 1. 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* (62) and *Pna2*_{1} (33) phases of K_{2}SeO_{4}. The number of formula units *per primitive unit cell* for the given structures are Z(G) = 4 and Z(H) = 12. 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.

Consider the group-subgroup related phase transition between monoclinic *C*2/*c* (15) and *P*2_{1}/*c* (14) phases of CaTiSiO_{5}. The number of formula units *per primitive unit cell* for the given structures are Z(G) = 2 and Z(H) = 4. A simple application of the formula gives an index 2 for this transformation.

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

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[ Check the t-subgroups of space group

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[ Check the k-subgroups of space group

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