# Homework 7: Algebra

Problem 1

Let $$A=\{1,2,3,4,5\}$$ and $$\circ$$ an operation on $$A$$ defined as $$a\circ b=a$$. Prove that $$\circ$$ is associative.

Problem 2

Assume that $$(G,\circ)$$ and that $$H\leq G$$ is its subgroup. If $$a\in G\setminus H$$, prove that $K=\left\{a^{-1}ha: h\in H\right\}$ is a subgroup of $$G$$.

Problem 3

Let $$p$$ be a prime number and $$\mathbb Z_p^*=\{ 1, 2, 3, \dots, p-1\}$$. For $$a,b\in \mathbb Z_p^*$$ let us define $$a\cdot b$$ to be the unique element $$z$$ of $$\mathbb Z_p^*$$ such that $$ab\equiv z$$ (mod $$p$$). Prove that $$\left(Z_p^*,\cdot\right)$$ is an abelian group.

Problem 4

Does there exist a non-commutative group of $$6$$ elements?

Problem 5

Let $$(R,+,\cdot)$$ be a ring with unity. If for all $$a,b\in R$$ we have $$(ab)^2=a^2b^2$$, prove that $$ab=ba$$ for all $$a,b\in R$$.

Problem 6

Assume that $$(G,\circ, e)$$ is an infinite group. Assume that $$f:G\to\mathbb R$$ is a function such that for every $$a\in G\setminus \{e\}$$ and every $$b\in G$$ we have $\max\left\{f(a\circ b), f\left(a^{-1}\circ b\right)\right\}> f(a).$ Assume that $$A$$ and $$B$$ are two non-empty finite subsets of $$G$$. Prove that there exists $$g\in G$$ for which there is exactly one pair $$(x,y)\in A\times B$$ that satisfies $$g=x\circ y$$.

Problem 7

Let $$(R,+,\cdot)$$ be a ring. If for each $$a\in R$$ we have $$a^3=a$$, prove that $$R$$ is commutative, i.e. $$a\cdot b=b\cdot a$$ for all $$a,b\in R$$.

Note: We say that $$(R,+,\cdot)$$ is a ring if $$(R,+)$$ is an abelian group, $$(R,\cdot)$$ is a semigroup, and for all $$a,b,c\in R$$ we have $$a\cdot (b+c)=a\cdot b+ a\cdot c$$ and $$(b+c)\cdot a=b\cdot a+ c\cdot a$$.

Problem 8

Let $$R$$ be a ring of characteristic $$0$$ (that is for every $$a\in R\setminus\{0\}$$ and every $$n\in\mathbb N$$ we have $$na\neq 0$$). Let $$e$$, $$g$$, and $$f$$ be idempotent elements of $$R$$ (element $$a$$ is idempotent if $$a^2=a$$). If $$e+g+f=0$$ prove that $$e=f=g=0$$.

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