Group I of the PT

Having only one valency electron, the atom / element belongs to group I; so also Hydrogen belongs there.
But sometimes writers have the tendency to put Hydrogen not in group I, just because of its special character. the speciality is that the one and only electron occurs in the one and only main level of Hydrogen. Normally, one electron in the outer shell is donated easily, but certainly not in the case of Hydrogen.
There are only two elements with only one main level, i.e. with only one sublevel 1s. They are:
Normally one electron in the outer shell is donated easily (and mostly we have then metals), but not in the case of Hydrogen. Het Hydrogen atom, after losing its electron, would be 'naked'. Only a nucleus of one proton would remain.
Well, that is impossible. protons do not exist independently. That's why Hydrogen is not e metal. All other elements in the first main group are (very reactive) metals, easily losing an electron.
Hydrogen atoms prefer to look for an electron, to gain an electron, just to have two electrons in the outer shell (what in this case is a noble gas structure). Hydrogen can do this only in cooperation with other atoms, sharing electrons with other atoms.
The result is a covalent bonding, but we will talk about that in module 03. In very exceptional cases Hydrogen can make a negative ion, the hydride ion H-.
H+ (the result of losing an electron) does not really exists, but very often you will find H+ in literature. It is just easy to use H+, to pretend that is exists.
All other elements of group I, Li to Fr, easily form positive ions, and these elements are very reactive.

The use of the symbol H+ in fact is kind of illegal.