Groups

Groups are one of the simplest and most prevalent algebraic objects in physics. Geometry, which forms the foundation of many physical models, is concerned with spaces and structures that are preserved under transformations of these spaces. At least one source of the prevalence of groups in physics is the fact that if these transformations are automorphisms, they naturally form a group under composition, called the automorphism group (AKA symmetry group).

The group of automorphisms of a set is called the symmetric group. For the finite set with \({n}\) elements, the elements of the symmetric group \({S_{n}}\) are called permutations, and any subgroup of \({S_{n}}\) is called a permutation group. The subgroup of all even permutations, i.e. permutations that are obtained by an even number of element exchanges, is called the alternating group \({A_{n}}\).

Some common group constructions include:

  • Normalizer of a subgroup \({H}\) of \({G}\): \({N(H)\equiv\left\{ n\in G\mid nHn^{-1}=H\right\} }\)
  • Center of a group: \({Z(G)\equiv\left\{ z\in G\mid zg=gz\;\forall g\in G\right\} }\)
  • Centralizer of a subgroup \({H}\) of \({G}\): \({C(H)\equiv\left\{ c\in G\mid chc^{-1}=h\;\forall h\in H\right\} }\)
  • Inner automorphism induced by \({a\in G}\): \({\phi_{a}(g)\equiv aga^{-1}}\)
  • Order of an element: \({|g|}\) is the smallest \({n}\) such that \({g^{n}=\mathbf{1}}\) (may be infinite)
  • Order of a group: \({|G|}\) is the number of elements in \({G}\) (may be infinite)
  • Torsion: \({\mathrm{Tor}(G)\equiv}\) elements of finite order; \({\mathrm{Tor}(G)}\) is a subgroup for abelian \({G}\)
  • Torsion-free: \({\mathrm{Tor}(G)=\mathbf{1}}\)

Some of the more important theorems about finite groups include:

  • Cayley’s theorem: every finite group is isomorphic to a group of permutations
  • Lagrange’s theorem: if \({H}\) is a subgroup of \({G}\), \({|H|}\) divides \({|G|}\)
  • Cauchy’s theorem: if \({p}\) is a prime that divides \({|G|}\) then \({G}\) has an element of order \({p}\)
  • The fundamental theorem of finite abelian groups: every finite abelian group can be uniquely written as the direct product of copies of the integers modulo prime powers, with the group operation applied component-wise; i.e. every finite abelian group is of the form
\(\displaystyle \mathbb{Z}_{p_{1}^{n_{1}}}\times\mathbb{Z}_{p_{2}^{n_{2}}}\times\dotsb\times\mathbb{Z}_{p_{k}^{n_{k}}}\)

where \({p_{i}}\) are not necessarily distinct primes

This last theorem has many consequences, including:

  • Any finitely generated abelian group can be written as above, but with some number of \({\mathbb{Z}}\) components also present
  • Any cyclic group (generated by a single element) is isomorphic to \({\mathbb{Z}_{n}}\)
  • \({|g|}\) always divides \({|G|}\); all groups of prime order are of the form \({\mathbb{Z}_{n}}\)

We do not discuss normal subgroups here; they will be covered in Quotient groups in the section Dividing algebraic objects.

An Illustrated Handbook