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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