Ramanujan's Summation says that the sum of all integers is -1/12... 1 + 2 + 3...=-1/12.
If we define group G to be group of all positive integers, then the group contains all positive integers. Since -1/12 is negative, and the group only contains positive integers, -1/12 is not an element of the group.
Therefore, Ramanujan's Summation is wrong.
On
Generalizations of the notion of "infinite sum" necessarily sacrifice some of the desirable algebraic properties of ordinary sums in order to attain greater generality. There's no "free lunch," so to speak.
The definition of "sum" under which the series you describe "sums" to $-\frac{1}{12}$ (clearly) does not have the property that a "sum" of positive numbers must be positive.
It's important to note that there's no "one true definition" of "sum," and that we may pick and choose which definition we are using for the purpose we're currently interested in. In some situations, it is useful to define "sum" such that Ramanujan's identity holds, even if it renders our notion of "sum" less ideal in other ways.
Your argument is invalid. To see why, consider the following:
“Ordinary summation says $$1-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+\ldots = \frac{\pi}{4}$$ But the rationals are closed under addition. Therefore ordinary summation is wrong.”
That a set is closed under addition (i.e. finite sums) does not imply it’s closed under summation (i.e. infinite sums).
Induction lets you prove properties of finite sums only.