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# abstract algebra代写|group thoery代写

Abstract algebra不算是一门简单的学科，这门学科在国内叫做抽象代数，经常有很多学生在学linear algebra或者analysis(advance calculus)的时候觉得并不困难，但是却觉得Abstract algebra很难，这是因为没有找到正确的方法学习Abstract algebra，UpriviateTA有一系列非常擅长Abstract algebra的老师，可以确保您在Abstract algebra取得满意的成绩。

identity The identity element is the function $$I: X \rightarrow G$$ which is identically equal to the identity element, $$e,$$ of $$G .$$ Indeed, for any $$f \in F$$ and any $$x \in X$$ we have $$(I * f)(x)=I(x) \cdot f(x)=e \cdot f(x)=f(x) .$$ Hence, $$I * f=f$$.
inverse Let $$f \in F$$ be any element of $$F .$$ Let $$g: X \rightarrow G$$ be defined by $$g(x):=$$ $$(f(x))^{-1}$$. Then for any $$x \in X$$ we have $$(g * f)(x)=g(x) \cdot f(x)=(f(x))^{-1}$$. $$f(x)=e=I(x) .$$ Hence, $$g * f=I$$ so that $$g$$ is a left-inverse of $$f$$.
associativity Let $$f, g,$$ and $$h$$ be elements of $$F$$. For any $$x \in X$$ we have $$f *(g * h)(x)=$$ $$f(x) \cdot(g * h)(x)=f(x) \cdot(g(x) \cdot h(x))=(f(x) \cdot g(x)) \cdot h(x)=(f * g)(x) \cdot h(x)=$$
$$(f * g) * h(x) .$$ Hence, $$f *(g * h)=(f * g) * h$$

$$a$$ must be the identity element.

to one. Thus, $$A$$ does not have an inverse in $$G$$.

$$x \mapsto\left\{\begin{array}{l} x+4 \text { if } x<12 \\ x-12 \text { if } x \geq 12 \end{array}\right.$$
Show that $$\sigma$$ is a permutation and describe its orbits.

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