More about subgroups and classes

If G and H are two groups for which a multiplication rule exists, that is to say the result g, h is defined, then the conjugate of H by an element g, G is 

When the result is H itself, H is invariant under the element g  

H are two groups for which a multiplication rule exists then the set of all the elements of G that commute with a particular element hj of H form a subgroup of G called the centralizer of hj in G, denoted by 

A class was defined in Section 1.2 as a complete set of conjugate elements. The sum of the members g/Zj),/ =1,2,. .., c, of the class Zj that contains the group element 

Proof The coset expansion eq. (13) shows that G gk is the DP set of zp) and gr, which means that G may be generated by multiplying each of the z members of zp in turn by each of the t members of gr. Therefore, gk in eq. (12) may be written as 

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