What is the criteria to place an element in a particular group?

and the criteria to place a period?

Answer: The criteria to place elements into groups and periods lies in their atomic number

Explanation:

As we know that period is the horizontal rows of elements in the periodic table and groups are vertical column of elements in the periodic table.

In a period we place those elements that have same number of shells and in a group those elements are placed that have different number of shells and the number of shells increases from top to bottom in a periodic table

For example

Sodium having symbol Na, atomic number of sodium is eleven it's electronic configuration is 1s2,2s2,2p6,3s1, next to it in a period is places potassium having symbol, K, it's electronic configuration is 1s2,2s2,2p6,3s2.

Sodium and potassium both r placed in a period not in a group because both of these have same outer most shell, only the number of electron in outer most shell varies.