Tag Archives: gr.group-theory

Let be an infinite field of characteristic and consider the subgroup of generated by the matrices and . We now consider the subgroup of consisting of matrices with all of their diagonal entries equal to . As a set, this … Continue reading →

In this post we’ll prove that is not free as a -module, although it is torsion-free. Recall that, in the finitely generated case, torsion-free modules are free, by the structure theorem. For the sake of contradiction, assume that is free. … Continue reading →

Let and be sets; we’ll prove that iff , following Andreas Blass’ argument here. Suppose and pick a bijection . Then is an isomorphism between and . Reciprocally, suppose . In the finite case, we can recover the cardinalities of … Continue reading →

If and are sets of the same cardinality, the group morphism that arises from a bijection is an isomorphism (its inverse is just the morphism induced by the bijection’s inverse). Conversely, we’ll prove that if and are not equinumerous sets, … Continue reading →

Let be a prime number and consider , the symmetric group on letters. Since and , the -Sylow subgroups of are cyclic of order , and they are generated by -cycles. An easy counting argument shows that there are exactly … Continue reading →