If element A belongs to the third period of the modern periodic table and is in group 2. Then what is the formula of the chloride of element A?

The group 2 elements are almost exclusively from ionic compounds containing $M^{2+}$ ions. 

In the modern periodic table:

• 'A' belongs to the third period because the periods are numbered sequentially, and the third period includes elements from sodium (Na) to argon (Ar).
• 'A' is in group 2, which is also known as the alkaline earth metals group. Group 2 elements have two valence electrons and share similar chemical properties.
• Element A is ${\mathit{MgCl}}_2$. 
  Magnesium (Mg) has an atomic number of 12, and its electronic configuration is [Ne] 3s², placing it in the third period and group 2 of the periodic table.

Magnesium chloride is an ionic compound. It is used in industrial, food, and medical industries.  Because of their higher ionisation energy and small size, both Be and Mg form organometallic compounds.
Valency of Mg = $2^{+}$

Valency of Cl   = $1^{-}$

Therefore, the formula of the chloride of element A is: ${\mathit{ACl}}_2$