Subgroup invariant

Suppose H is an invariant subgroup of a group G and that HX. HP.. .. arc its eosets. The elements of If can he considered collectively as the unit element of another group and the various covets as the remaining elements. 

Exercise 1.3-1 Prove that any subgroup of index 2 is an invariant subgroup. 

Consequently, Hi is not an invariant subgroup. For H to be an invariant subgroup of G, right and left cosets must be equal for each coset representative in the expansion of G. 

Suppose that H is an invariant subgroup of G of index t. Then the t cosets gr H of H (including y, H = H) each considered as one element, form a group of order t called the... 

Note that in semidirect products the invariant subgroup is always the first group in the product. For example,... 

The second method is applicable to proper point groups P that have an invariant subgroup Q of index 2, so that... 

D2 has the invariant subgroup D of index 2, with the coset expansion... 

Table 2.8. The relation of the point groups 0 and T dto their invariant subgroup T.

Magnetic crystals with a net magnetic moment belong to one of the point groups G which contain complementary operators, but for which 0 < G. If G = H + QH, where H is a halving subgroup (invariant subgroup of index 2) of G, and Q C G but Q H, then... 

Consider the group G = H, AH that contains unitary H and antiunitary AH operators. H is necessarily an invariant subgroup of G of index 2 and AH is a coset of H with coset representative A (which may be any one of the antiunitary operators of G) so that... 

Equation (11) shows that the set of lattice translations T form an Abelian subgroup of G. Moreover, T is an invariant subgroup of G, since... 

The decomposition of r(NCf) JF modulo the invariant subgroup r(NCf) g defines a factor group isomorphic to the internal isometric groups (I)... 

---