Putnam preparation (Table of contents)

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 \(e=f=g=0\).