Centralizer of a subgroup

 Centralizer of a Subgroup:

 Let be a subgroup of The set of those elements of which commute with every element 

of is called centralizer of in Mathematically, it is written as:{g∈G:gh=hg for all h∈H}

 

Note: Centralizer can also be defined for any subset of a group 

1 CG(H) is non-empty

2 CG (H) is a subgroup of G for any subgroup H of G

3 if G be an ablian group CG(H)=G

4 The centerlizer of a subgroup need not contain that sub group

5 The Centralizer of the whole group is the center of the group

6 The number of element in a conjugacy class Ca of an element a∈G is equal to Index of it's normalizer in G thus |Ca|=|G:CG(a)|.