Free Groups and Group Presentations

Is the group of positive rational numbers under multiplication torsion-free? Is it a free abelian group?

Let \(M\) be a \(\mathbb {Z}\)-module consisting of \(3\times 1\)-matrics with integer entries. Let \(N\subseteq M\) be a \(\mathbb {Z}\)-submodule generated by the columns of the matrix \[ \begin {pmatrix} 2 & 7 & 3 & 4\\ 3 & 8 & 7 & 11\\ 4 & 10 & 10 & 16 \end {pmatrix}. \] Find a minimal presentation for the quotient \(M/N\).

Show that a free group \(F\) is torsion-free; that is, that if \(g\ne 1\) is an element of \(F\), then \(g^n\ne 1\) for all \(n\ge 1\).

Prove that a free group has no nonidentity elements of finite order.

Let \(f\colon \mathbb {Z}^n\to \mathbb {Z}^n\) be a map of free Abelian groups of rank \(n\) defined by a matrix \(A\) with non-zero determinant. Show that the cokernel of \(f\) is finite and that its order is equal to the absolute value of the determinant of \(A\).

Explain what a free group is and prove that any two free bases for the free group \(F\) have the same cardinality.

Prove that the group given by the presentation \[ \langle x,y\mid x^2=1,\ y^2=1,\ xyx=yxy\rangle \] has 6 elements.