The Abstract Groups

Table 16.9. Character table of the abstract group Gg, o/ t/ze covering group Cf and its isomorph D4. showing the corresponding classes ...

Wi = Rn and W2 = Ri2, where Rn, Ri2 are representations of the abstract group G32 which is isomorphous with + (k). Column headings for the classes are the coset representatives ( A w). Time-reversal symmetry is of type a for both representations. 

XT(k) is isomorphous with the abstract group G32 with generators... 

The symbol J ( ) will henceforward be used as the abstract group JF iX) =... 

Wussing, H. 1984. Genesis of the Abstract Group Concept (translated by Shenitzer, A.). Cambridge, MA MIT Press. 

As for the inverse of each element, an examination of Table II shows that / is its own inverse, C z) is its own inverse, and likewise a,(xz) and (Ti(yz) are their own inverses. Hence, the set of geometric operations that has the equilibrium configuration of H2O as its invariant forms a group. These considerations can be generalized to define the abstract group G. By the abstract group G is meant a set of elements a, 6, c,... with a binary operation defined such that the following axioms are obeyed. 

The two examples cited are usually referred to as realizations of the abstract group. 

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