0.5 Group theory

If $G$ is a group then a subgroup $H$ is a subset containing the identity, closed under the group law and taking inverses. If $H$ is a subgroup, then a left coset of $H$ in $G$ is a subset $gH=\{gh:h\in H\}$. The left cosets partition $G$, and we write $G/H$ for the set of left cosets (not a group!). Similarly we define right cosets and $H\backslash G$.

A subgroup $H$ is normal if, for every $g\in G$, $gH{g}^{-1}=H$. Equivalently, $gH=Hg$ for all $g\in G$. In this case, the rule

$$(gH)(g^{\prime }H)=(g{g}^{\prime })H$$

defines a group law on $G/H=H\backslash G$ (the quotient group).

If $G$ and $G^{\prime }$ are groups, a homomorphism $f:G\to G^{\prime }$ is a function such that $f(gh)=f(g)f(h)$. The kernel of $f$ is $\ker (f)=\{g\in G:f(g)=e\}$ and the image is $\mathrm{im}(f)=\{f(g):g\in G\}$ and these are subgroups of $G$ and $H$ respectively. The subgroup $\ker (f)$ is normal — in fact, normal subgroups are precisely those that are the kernel of some homomorphism.

A homomorphism is injective if and only if its kernel is trivial. A bijective homomorphism is called an isomorphism, in which case the inverse is also an isomorphism and we say that the groups are isomorphic.

The first (and best) isomorphism theorem states that the map

	$G/\ker (f)$	$\to \mathrm{im}(f)$
	$g\ker (f)$	$\mapsto f(g)$

is an isomorphism.

The symmetric group $S_n$ is the group of permutations of $\{1,\mathrm{\dots },n\}$. We use cycle notation, so that (e.g.) $(1253)$ is the permutation taking $1$ to $2$, $2$ to $5$, $5$ to $3$, and $3$ to $1$. Every permutation can be written uniquely (up to changing the order of the factors) as a product of disjoint cycles. We don’t bother writing cycles of length one, for example

$$(18)(2475)\in S_8$$

fixes the elements $3$ and $6$. The sequence of lengths of the cycles appearing (including those of length one!), written in decreasing order, is called the cycle type of the permutation. The elements of given cycle type make up a single conjugacy class of $S_n$ (see below for conjugacy classes).

There is a homomorphism $ϵ:S_n\to \pm 1$ uniquely determined by the property that it takes transpositions $(ij)$ to $--1$. It is called the sign homomorphism. Its kernel is the alternating group $A_n$. The sign of an $n$–cycle is ${(-1)}^{n+1}$.

If $X$ is a set, we sometimes write $S_X$ for the group of permutations of $X$, so $S_n=S_X$ with $X=\{1,\mathrm{\dots },n\}$. If $|X|=n$ then $S_X\cong S_n$, but the exact isomorphism depends on how we label the elements of $X$ by the numbers $1$ to $n$.

A (left) action of a group $G$ on a set $X$ is a way of transforming an element $x$ by elements $g$ to produce $gx\in X$, such that $(gh)x=g(hx)$.

In this case, if $x\in X$ then its stabiliser $\mathrm{stab}(x)=\{g\in G:gx=x\}$ is a subgroup. We also have the orbit $Gx=\mathrm{orb}x=\{gx:g\in G\}\subset X$. One form of the orbit-stabiliser theorem states that the map

	$G/\mathrm{stab}(x)$	$\to Gx$
	$g\mathrm{stab}(x)\mapsto gx$

is a bijection. This implies (if $G$ is finite) that

$$|\mathrm{orb}x||\mathrm{stab}(x)|=|G|.$$

If $g\in G$ then the set of fixed points of $g$ is

$$X^g=\mathrm{Fix}(g)=\{x\in X:gx=x\}.$$

We write

$$X^G=\{x\in X:gx=x\text{ for all }g\in G\},$$

the set of fixed points of $G$.

We will need a few examples of groups:

• •

  the integers $\mathbb{Z}$

• •

  cyclic groups $C_n\cong \mathbb{Z}/n\mathbb{Z}$

• •

  symmetric groups $S_n$

• •

  alternating groups $A_n$

• •

  dihedral groups $D_n$ (symmetries of regular $n$–gon)

• •

  the quaternion group $Q_8$:

  $$\{\pm 1,\pm i,\pm j,\pm k\}$$

  with $i^2=j^2=k^2=ijk=-1$.

• •

  the general linear group ${\mathrm{GL}}_n(k)$ (for $k$ a field) of invertible $n\times n$ matrices over $k$ [note that if $k={𝔽}_p$ then this is finite!]

• •

  the special linear group ${\mathrm{SL}}_n(k)$ of matrices with determinant 1

• •

  the orthogonal group

  $$\mathrm{O}(n)=\{A\in {\mathrm{GL}}_n(ℝ):A{A}^T=I\},$$

  which is also the group of isometries of ${ℝ}^n$ (with its standard inner product) fixing the origin, and the subgroup $\mathrm{SO}(n)$ of elements of $\mathrm{O}(n)$ whose determinant is one (i.e. the group of rotations of ${ℝ}^n$ fixing the origin.)

• •

  the unitary and special unitary groups

  $$\mathrm{U}(n)=\{A\in {\mathrm{GL}}_n(ℂ):A{A}^{†}=I\}$$

  where $A^{†}={\overline{A}}^T$, with $\overline{\cdot }$ being complex conjugation, and

  $$\mathrm{SU}(n)=\{A\in \mathrm{U}(n):det(A)=1\}.$$

  The group $\mathrm{U}(n)$ is the group of transformations of ${ℂ}^n$ fixing the standard Hermitian inner product.