# 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# isconjugatewith#Q# , meaning it is in thesame classas#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, *in its own class* and thus is nothing like any other element in the group.