In mathematics, the Dedekind eta function, named after Richard Dedekind, is a modular form of weight 1/2 and is a function defined on the upper half-plane of complex numbers, where the imaginary part is positive.
== Definition ==
For any such complex number τ, let q = exp(2πiτ), and define the eta function by,
The notation is now standard in number theory, though many older books use q for the nome . Its 24th power gives,
where Δ is the modular discriminant. The presence of 24 can be understood by connection with other occurrences, such as in the 24-dimensional Leech lattice.
The eta function...
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A subset of a group doesn't necessarily have closure. To show closure, you need to select two arbitrary elements of $A$, say $x, y \in A$, and show that $xy \in A$.
Let $x, y \in A$. Then $x^n = e, y^n \in e$, where $n\in \mathbb Z$. Then $x^ny^n = ee = e = (xy)^n$, because $G$ is abelian. Hence...
