/sci/ - Science & Math

>International seminar for young researchers "Algebraic, combinatorial and toric topology"
>(December 17–18, 2020, online)
http://www.mathnet.ru/php/conference.phtml?confid=1885&option_lang=eng
Moscow time GMT+3!
>>12472937
I'm not sure if I've ever done that.

>>12202252
Basically, you just take (x+y)^n and forget all the coefficients. To address your points:
(2) every element of a free group will be a word consisting of the generators. If we know how the generators are mapped, then we can define a new function using that information. For example, let's go with your idea for the 3 generators. We have a function [math]f \colon \{ x, y, z \} \to \{ a, b\}, x \mapsto a^2, y \mapsto ab, z \mapsto b^2[/math], and we want to use this to define a function [math]\varphi\colon F(x, y, z) \to F(a, b)[/math]. We do this the following way: [math] \varphi( x^{k_1} y^{m_1} z^{n_1} x^{k_2} \cdots x^{k_p} y^{m_p} z^{n_p}) = f(x)^{k_1} f(y)^{m_1} \cdots f(z)^{n_p}[/math] (notice that the function [math]f[/math] can actually be anything here). This gives a well-defined function.
(1) Using the thing above, it is easy to see that [math]\varphi (gh) = \varphi(g) \varphi(h)[/math] for all elements.