Topological group

A topological group is a group whose multiplication and inversion maps are continuous.

A compact topological group is a topological group whose underlying topological space is compact and Hausdorff.

A profinite group is an inverse limit of finite discrete groups. Equivalently, it is a compact Hausdorff totally disconnected topological group.

A profinite abelian group is an inverse limit of finite abelian groups. Its topology is compact and totally disconnected. An abelian pro-p group acquires scalar multiplication by p-adic integers by taking limits of integer multiples.

A pro-$p$ group is an inverse limit of finite $p$-groups. Every continuous action on a finite set or finite module factors through a finite $p$-group. In particular its action on ${\mu }_p$ is trivial when the field characteristic differs from $p$, because $\mathrm{Aut}({\mu }_p)$ has order $p-1$.

A Sylow pro-$p$ subgroup of a profinite group is a closed pro-$p$ subgroup whose images are Sylow $p$-subgroups in every finite quotient. Such subgroups exist and are conjugate. Every open subgroup containing one has index prime to $p$. They allow prime-to-$p$ finite extensions to detect $p$-primary Galois cohomology.

The profinite topology on a group takes the cosets of finite-index normal subgroups as a neighborhood basis. A profinite group carries the inverse-limit topology from its finite quotients.

An inverse limit carries the coarsest topology making every coordinate projection map continuous. It is the subspace topology inherited from the product of the component spaces.

The profinite completion of a group $G$ is the inverse limit of its finite quotient groups:The canonical map $G\to \hat{G}$ has dense image and is injective exactly when $G$ is residually finite.

Every homomorphism from a group $G$ to a profinite group $P$ extends uniquely to a continuous homomorphism $\hat{G}\to P$. Construct the extension on every finite quotient of $P$ and invoke the universal property of an inverse limit; uniqueness follows from the density of $G$ in $\hat{G}$.

A subset $S$ of a topological group topologically generates it when the ordinary subgroup $⟨S⟩$ is dense.

A topological group is topologically finitely generated when it has a finite topological generating set.

For a profinite group $G={\underleftarrow{\lim }}_j{G}_j$, a subset $S\subseteq G$ is a topological generating set exactly when $⟨p_j(S)⟩=p_j(G)$ in every finite quotient. Indeed, a subgroup of a profinite group is dense exactly when its image in every finite continuous quotient is surjective.

Elements $g,h$ of a profinite group $G={\underleftarrow{\lim }}_j{G}_j$ are conjugate exactly when their images are conjugate in every $G_j$. For the nontrivial direction, the finite nonempty sets of conjugators in the $G_j$ form an inverse system, whose inverse limit supplies a conjugator in $G$.

A topological group has the topological Hopf property when every continuous surjective endomorphism is a topological automorphism.

Every topologically finitely generated group that is profinite has the topological Hopf property. There are only finitely many open subgroups of each given index, and inverse image under a surjective endomorphism permutes them. The endomorphism kernel consequently lies in every open normal subgroup and is trivial.

For $G={\underleftarrow{\lim }}_j{G}_j$, the equation $x^m=g$ has a solution in $G$ exactly when it has a solution after projection to every $G_j$. The finite nonempty sets of roots form an inverse system, and a compatible family is a root in $G$.

For loops based at the identity of a topological group, pointwise multiplication and concatenation obey the interchange law and have the same identity. The Eckmann-Hilton argument therefore makes the fundamental group abelian.

If two unital operations on one set share their identity and satisfy $(a\ast b)\cdot (c\ast d)=(a\cdot c)\ast (b\cdot d)$, then the operations coincide and are commutative.

For $n\ge 1$, pointwise multiplication of based cube maps into a topological group induces the ordinary addition in its homotopy group ${\pi }_n(G,e)$. Concatenation and pointwise multiplication share an identity and satisfy the interchange rule, so the Eckmann-Hilton argument identifies the two operations. In particular, the $m$-th power map induces multiplication by $m$ on every homotopy group, for all integers $m$.

The unitary group $U(n)$ consists of complex matrices preserving the standard Hermitian inner product.

The subgroup $\{U\in U(2n):UJ{U}^T=J\}$, with $J=\left(\begin{array}{cc}0& I_n\\ -I_n& 0\end{array}\right)$, is the compact symplectic group, often denoted $Sp(n)$ or $USp(2n)$. It preserves both a positive Hermitian metric and a nondegenerate complex alternating form.

The matrix logarithm near the identity maps $Sp(n)=\{A\in U(2n):AJ{A}^t=J\}$ onto an open set of its real compact symplectic Lie algebra $\{X:X^{\ast }=-X,XJ+J{X}^t=0\}$. Its inverse is the matrix exponential. These statements follow by applying the logarithm to $A^{\ast }=A^{-1}$ and $A^t=J^{-1}{A}^{-1}J$, and differentiating $e^{tX}J{e}^{t{X}^t}$. Left translations provide manifold charts at every group element. In block form $X=\left(\begin{array}{cc}P& Q\\ -\bar{Q}& \bar{P}\end{array}\right)$ with $P^{\ast }=-P$ and $Q^t=Q$, giving real dimension $n^2+n(n+1)=n(2n+1)$.

The special unitary group is $SU(n)=\{U\in U(n):\det U=1\}$.

The four-by-four unitary matrices of determinant one form this special unitary group. Its two-index antisymmetric representation has complex dimension six and the real structure of the SU4 exterior square realizes it on a six-dimensional real space.

For $n\ge 2$, the special unitary group acts transitively on the complex unit sphere. Extend any unit vector to an orthonormal basis, then adjust the phase of a remaining column to give determinant one. The stabilizer subgroup of the first vector consists of special unitary transformations of its orthogonal complement. Thus the sphere is a homogeneous quotient, not just a set with a norm-preserving action.

The SU(3) group consists of unitary complex three-by-three matrices of determinant one. Its defining action on ${\mathbb{C}}^3$ preserves the Hermitian norm. It contains block-diagonal copies of SU(2) and the real SO(3) group. Its Lie algebra consists of traceless anti-Hermitian matrices.

The SU(2) group is the group of two-by-two unitary matrices of determinant one. It is a compact three-dimensional Lie group whose underlying manifold is a three-sphere.

An SU(2) matrix is a two-by-two complex unitary matrix with determinant one. It has the form $\left(\begin{array}{cc}a& b\\ -\bar{b}& \bar{a}\end{array}\right)$ with $|a{|}^2+|b{|}^2=1$. Its action on the Riemann sphere is a rotational Möbius transformation; opposite matrices induce the same sphere rotation.

Every element of $SU(2)$ has the unique formThus $SU(2)$ is diffeomorphic to the unit three-sphere in ${\mathbb{C}}^2$ and is a parallelizable manifold.

The determinant $U(n)\to S^1$ sends a unitary loop to a circle-valued loop. A nonzero determinant winding number proves that the original loop class has infinite order.