Group theory 5: products

This lecture, part of an online group theory course, focuses on products of groups. It begins by classifying groups of order 4, showing that they are either cyclic (Z/4Z) or the Klein four-group (Z/2Z × Z/2Z). The classification uses Lagrange’s theorem and the property that if all elements have order 2, the group is abelian and can be viewed as a vector space over F2. The lecture then introduces exact sequences, distinguishing split and non-split extensions, and warns against assuming all exact sequences split. Several examples of products are given: the multiplicative group of non-zero reals as a product of {±1} and positive reals, complex numbers as a product of the circle group and positive reals, and the Chinese remainder theorem for groups. The lecture also discusses the structure of the multiplicative group of units modulo mn, showing that Z/15Z* is isomorphic to Z/4Z × Z/2Z, which sharpens Euler’s theorem. Further examples include the non-zero rationals as a direct sum of copies of Z and a group of order 2, and the symmetry groups of Platonic solids, where the octahedron, cube, icosahedron, and dodecahedron split as products but the tetrahedron does not. Finally, the group of all roots of unity is shown to be a direct sum of prime-power root subgroups.