Right coset

These subgroups of G generated by multiplication of some element of G by the subgroup H are called cosets of H in G. If the multiplication is carried out on the left, they are called left cosets, and vice versa for right multiplication. 

We conclude that one can find a number of right coset generators giving distinct cosets until the permutations of S are exhausted. S5mibolizing the right coset generators as ti = /, T2,. .., r, we have... 

Our goal now is to find a convenient set of right coset generators for Gj that gives S . Let us now consider specifically the case for the 2 partition with... 

Note that the elements of G do not have to appear in exactly the same order in the left and right coset expansions. This will only be so if the coset representatives commute with every element of H. All that is necessary is that the two lists of elements evaluated from the coset expansions both contain each element of G once only. It should be clear from eqs. (5) and (6) that H gr = gr H, where H= P0 Pi P2 and gr is P4. An alternative way of testing for invariance is to evaluate the transforms of H. For example,... 

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. 

Now consider the symmetry point group G (or, more precisely, the framework group ) of the above ML coordination compound. This group has IGI operations of which lf l are proper rotations so that IGI/I/ I = 2if the compound is achiral and IGI/I I = 1 if the compound is chiral (i.e., has no improper rotations). The n distinct permutations of the n sites in the coordination compound or cluster are divided into nM R right cosets which represent the permutational isomers since the permutations corresponding to the IWI proper rotations of a given isomer do not change the isomer but merely rotate it in space. This leads naturally to the concept of isomer count, I, namely,... 

Configurations equivalent to s, under figure index symmetry constitute the right coset of the isomorphic image of P with respect to s,... 

Prove that two right (or left) cosets of a subgroup are either identical, in that they are the same set of elements, or every element of one differs from every element of the other. 

A,U is called a left coset of U. The right coset of U is UA, never contain the identity element. 

The proof illustrates the connection between cosets and conjugacy classes. A special example of this arises in the case of normal, or invariant, subgroups. A subgroup H is normal if its left and right cosets coincide, i.e., if RiH = HRi. This implies that aU the elements of the group will map the subgroup onto itself or, for a normal subgroup H,... 

Here g is a coset generator outside H. The subgroup is normal if the right and left cosets coincide. Since there is only one coset outside //, it is required that... 

Definition (Cosets and double cosets) Consider a group G together with a subgroup A < G. Then we have the following actions of A on G, from the left and from the right, respectively ... 

SALWFs are partitioned into subsets (/) belonging to a class b, each transforming into itself under the operators of a local symmetry point-subgroup F/ of the full point group F, while the operations of the associated left- or right-cosets transform subset / into another equivalent one in the same class. As an example in the diamond crystal... 

Ci(y)[Pg.197]

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