We should be careful when defining "properties" - chemical properties are probably the ones that an instructor would be taking into consideration. By contrast, elements of the same group have very different numbers of protons, masses, spectral lines, and abundances in the universe, so we cannot necessarily equate groups with overall similarity among elements.
Chemical properties generally refer to how we predict an element will react with other matter. Valence electrons are the primary determinor of chemical reactions, and so we would expect atoms with the same number of valence electrons to undergo the same kinds of reactions, such as bonding with element X but not with Y, or releasing an electron in the presence of A but not B.
The connection here is that groups on the periodic table are organized according to valence electrons; all elements in a group have the same number of valence electrons. However, even here we have to be cautious with generalizations; just because Hydrogen and Lithium have one valence electron doesn't mean those electrons are the same, and will react the same. For example, the higher the element number, the farther away the electron is. This will result in similar reactions, but the strengths of those reactions will be considerably different.
Atoms of one element react with atoms of other elements to change the number of electrons in the outermost shell. The objective of the reactions is to achieve a stable configuration which is that of a noble gas.
Elements of the same group have the same kind of electronic configuration. The number of electrons in the outermost shell is the same. So the elements have to gain or lose the same number of electrons. This makes their properties similar.
For example elements in the alkali metal group have one extra electron in their outermost shell. If this electron is lost they form an ion with a single positive charge and the electronic configuration changes to that of a noble gas. So, you will notice that all elements of this group form compounds where they are positive ions with a single charge.
Electrons determine the chemical reactivity of an element, so the elements in the same group have the same number of electrons on the outer post shell, giving them similar chemical properties.