For main-group elements using the older numbering system the group number on the periodic table corresponds to which characteristic of the elements of that group

The Group number and period number corresponds to the outer most shell and number of electrons present in outer most shell.

For Example:

Oxygen has atomic number 8, and its electronic configurations is,

1s², 2s², 2pₓ², 2py¹, 2pz¹

The outermost shell is 2, means oxygen is present in period 2, and the number of electrons in n = 2 are 6, so the group number is 6 in periodic table.

Lets take another example of Sodium, it has atomic number 11 with electronic configuration;

1s², 2s², 2p⁶, 3s¹

The outermost shell is 3, means sodium is present in period 3, and the number of electrons in n = 3 are 1, so the group number is 1 in periodic table.