Group Theory

1 Basics

Groups are an algebraic theory with \(1\) nullary (denoted \(1\)), unary (denoted \(^{-1}\)), and binary (denoted \(*\)) operation, such that \(*\) is associative, \(^{-1}\) is a left and right inverse with respect to \(*\), and \(1\) is a left and right identity with respect to \(*\). One sometimes uses additive notation \(0,-,+\) instead of \(1,^{-1},*\), and also sometimes leaves out the \(*\). We can consider the category of models of this theory in \(\Set \) to get the category of groups, \(\Grp \). This category is complete and cocomplete, with product \(\times \) called the

direct product, coproduct \(*\) called the free product. A group is nothing more than a one object category where every arrow is an isomorphism, thus certain concepts for categories and groups coincide.

As is the case for any algebraic theory, we have facts such as a bijective group homomorphism is an isomorphism. Some remarks about groups: they can be seen as semigroups with an identity element and an inverse. Indeed, if a left and right identity exists, it is unique, and likewise for inverses. Every group is naturally isomorphic to its opposite group via the map \(x \to x^{-1}\).

Here is the main example of a group. Given an object \(O\) in a category, we can consider its automorphisms \(\Aut (O)\), namely the isomorphisms from it to itself, with the operation as composition. Now a group \(G\) can act on one of these objects. One distinguished between left and right actions by saying that a right action is an action of the opposite group. In particular, an action is a just a homomorphism \(\phi \) from \(G\) to \(\Aut (O)\). Generally when one says group action, \(O\) is a set, however another important example is when \(O\) is a vector space, in which case an action is called a representation.

There is some standard terminology when discussing actions. An action is faithful if the corresponding functor is faithful. For an action on a set, the orbit of an element is the set of elements that the group elements send it to. In particular, we say the action is transitive if there is one orbit. The fixed points of an action \(\phi \) is the equalizer of the image of \(G\). The stabilizer of a morphism \(h:B \to O\) (denoted \(S_\phi (h)\)) is the subgroup of \(G\) such that for each element \(g\) in the subgroup, \(g \circ h = h\).

Transitive set actions are extremely simple, and in fact (up to a point) correspond to subgroups of \(G\).

There are two set actions of a group \(G\) on itself we will now consider. First, the group acts on itself by left multiplication. This action is both faithful and transitive, so we say it is simply transitive. In particular, if \(S_{|G|}\) is the automorphism group of \(G\), which we call the symmetric group on \(|G|\) elements, then we get Cayley’s theorem.

Proposition 1.1 (Cayley’s Theorem). Every group is a subgroup of \(S_{|G|}\) which in particular is a finite symmetric group if \(G\) is finite.

Every subgroup \(H\) acts on \(G\) by left or right multiplication, and the quotient space is called the set of left (denoted \(H\backslash G\)) or right (denoted \(G/H\)) cosets. We often denote the coset a particular element \(b\) is in by \([b]\). \(G\) acts transitively on the cosets via left or right multiplication. In fact, by varying \(H\), we actually get all of the transitive actions. This is the orbit stabilizer theorem.

Proposition 1.2 (Orbit-Stabilizer Theorem). The category of pointed transitive set actions is equivalent to the category of subgroups of \(G\).

Proof. In one direction, we take a pointed transitive action to the stabilizer of the point, and in the other direction we take a subgroup of \(G\) and consider the left action on \(G/H\). It is an easy exercise to verify this is an equivalence, and see what the functors do on maps. □

Given a group action \(G\), and a (usually injective) map \(H\to G\), we can consider let \(H\) act via restriction. Restriction has an adjoint called induction. Restricting to \(H\) partitions the action into new orbits, and to see what the new orbit of a point \(p\) is, note its stabilizer is just \(S(p)\cap H\). We can translate this into the language of groups as follows: Fixing a base-point, our action becomes the right action of \(G\) on \(G/K\) some subgroup \(K\), and the orbits become double cosets, denoted \(HgK\), and each double coset breaks up into disjoint right cosets of \(K\). We can compute the size of the double coset of \(x\) to be \(|K||H|\frac {1}{|K\cap xHx^{-1}|}\) under the assumption that everything is finite.

The second important action of \(G\) on itself is actually an action as a group, called conjugation. In particular, \(g\) goes to the automorphism of \(G\) sending \(h\) to \(ghg^{-1} = h^g\). The fixed points of the action is a subgroup called the center, denoted \(Z(G)\). One says that \(G\) is abelian if \(Z(G) = G\). For the conjugation action, the orbits are called conjugacy classes (denoted \(Cl(a)\), and the stabilizers are called centralizers, denoted \(C_G(S)\) where \(S\) is a subset.

Additionally, \(G\) acts on its set of subgroups by conjugation, and the stabilizers here are called the normalizers, and denoted \(N_G(H)\).

Definition 1.3. A subgroup \(H\) of \(G\) is normal (denoted \(H \triangleleft G\)) if \(H\) as a set is fixed by conjugation (we denote this \(gHg^{-1} = H, \forall g \in G\)). One example of a normal subgroup is the center. Another example is the kernel of a homomorphism, ie. all the elements that are sent to the identity. In fact, as we will see below, when \(H\) is normal, \(G/H\) can be made into a group such that the projection \(G \to G/H\) is a homomorphism with kernel \(H\).

Proposition 1.4. The following are equivalent:

1. \(H\) is normal

2. \(G/H = H\backslash G\) as quotients of \(G\).

3. The operation \(*\) descends to \(G/H\).

Proof. \(1 \implies 2\): If \(H\) is normal, then \(Hc\)= \(cHc^{-1}c = cH\). \(2 \implies 3\): We define \(cH*dH = cdH\), which is a group since \(cHdH = cdHH = cdH\) as sets. \(3 \implies 1\): \(H\) is the kernel of the natural projection \(G \to G/H\) so is normal. □

Another important property of kernels is in the next lemma.

Lemma 1.5. A group homomorphism is injective iff the kernel is trivial.

Proof. Certainly a homomorphism is injective if the kernel is trivial. Conversely if the kernel is trivial and \(\phi (g) = \phi (h)\), then \(\phi (gh^{-1})\) is in the kernel so \(g = h\). □

Now we will prove the universal property of quotients.

Proposition 1.6 (Universal property of quotients). If \(G \to G'\) is a homomorphism with \(H\) in the kernel, then it uniquely factors through \(G/H\) as in the diagram

Proof. If we had such a homomorphism \(\gamma \), \(\gamma (cH) = \phi (c)\), so it is unique. Moreover, it is easy to see that this is a well-defined homomorphism since \(H\) is in the kernel and is normal. □

Corollary 1.7 (First Isomorphism Theorem). The induced map \(G/\ker (\phi )\to \im (\phi )\) is an isomorphism.

Proof. Indeed the kernel of the induced map is trivial by definition so the map is injective, moreover it is surjective by definition of the image. □

Corollary 1.8 (Second Isomorphism Theorem). If \(H,N\) are subgroup of \(G\), \(N\) normal, there is a canonical isomorphism \(HN/N \cong H/(N\cap H)\).

Proof. The canonical map \(H \to HN/H\) is surjective with kernel \(N \cap H\). □

Corollary 1.9 (Third Isomorphism Theorem). If \(H,N\) are normal in \(G\), \(H \subset N\), then \((G/H)/(N/H) \cong G/N\)

Proof. The projection \(G \to G/H \to (G/H)/(N/H)\) is surjective with kernel \(N\). □

Corollary 1.10 (Lattice Isomorphism Theorem). There is an isomorphism between the lattice of subgroups containing \(N\), a normal subgroup of \(G\), and the subgroups of \(G/N\). Moreover normal subgroups correspond to normal subgroups.

Proof. Given a subgroup containing \(N\), project it to \(G/N\), and conversely given a subgroup of \(G/N\), take its preimage. Then it is straightforward to see these are inverses that preserve the lattice structure. □