First member of a group differs from the rest of the members of the same group. Why?
1 Answer
I interpreted this to be Group Theory... of course, it's up in the air, and could be about the periodic table.
However, elements in a single periodic table group should be quite similar, except for mainly their atomic number, and possibly their phase at
The four postulates of a group are (given a group operation
- There exists an identity element
#E# in the group, such that for a given other element#A# in the group,#A@E = E@A = A# . - There exists an inverse element
#B# in the group such that#A@B = B@A = E# . - The group operation
#@# is associative, i.e.#A@(B@C) = (A@B)@C = A@B@C# . - Any two members
#A# and#D# in the group via the operation#A@D# generate another element#F# also in the group. We call this the "closure property".
For example, water belongs to the point group
The first element in a group is generally considered the identity element
The previous statement can be shown by performing a similarity transform on
If
#P^(-1)@Q@P = R# , then#R# is conjugate with#Q# , meaning it is in the same class as#Q# . Thus, if#R = Q# , then#Q# is in its own class.
And using the first postulate of a group,
#P^(-1)@(E@P) = P^(-1)@P = E#
#P^(-1)@(E@P) = (P^(-1)@P)@E = E@E = E#
Either way you approach the similarity transformation,