Group

A group is a set with an associative binary operation, an identity, and an inverse for every element.

The group of braids on $n$ strands, composed by stacking, with adjacent-crossing generators ${\sigma }_i$ and the braid group relations. It retains over/under crossings and is not a permutation group.

The generators ${\sigma }_i$ commute when $|i-j|>1$ and satisfy ${\sigma }_i{\sigma }_{i+1}{\sigma }_i={\sigma }_{i+1}{\sigma }_i{\sigma }_{i+1}$. Unlike a symmetric-group presentation, there is no ${\sigma }_i^2=1$ relation.

An infinite group is a group with infinitely many elements. An element of infinite order proves that its group is infinite, but the converse need not hold: an infinite group can have all elements of finite order.

The trivial group has exactly one element, its identity. Every group homomorphism to it is trivial, and a group is trivial precisely when all elements in a generating set equal the identity.

If every element of a group squares to the identity, every element equals its inverse. Then $ab=(ab{)}^{-1}=b^{-1}{a}^{-1}=ba$, so the group is an abelian group. The argument does not require a finite group.

A non-abelian group is a group containing elements $a,b$ for which $ab\ne ba$.

A torsion-free group is a group in which the identity element is the only element of finite order of a group element.

A divisible group is an abelian group $G$ in which, for every positive integer $n$ and every $g\in G$, some $h\in G$ satisfies $nh=g$.

An abelian group $A$ is $p$-divisible when multiplication by $p$ is surjective, equivalently $A/pA=0$. This does not mean it has no $p$-torsion: $\mathbb{Q}/\mathbb{Z}$ is divisible and has elements of every finite order. The Kummer cohomology divisibility criterion relates this property to Galois cohomology.

A divisible Abelian group is torsion-free precisely when division by each nonzero integer is unique. Defining rational scalar multiplication by $n((m/n)a)=ma$ identifies these groups with vector spaces over the rational numbers, including the zero space.

The group axioms require closure and associativity of the binary operation, an identity element, and an inverse for every element.

A finite group is a group with finitely many elements. Its number of elements is its order.

Every finite group whose order is divisible by at most two distinct primes is a solvable group. A nonabelian simple group must therefore have at least three distinct prime divisors.

A nonabelian simple group has no nonidentity conjugacy class of prime-power size. Coprime-degree irreducible characters vanish on such an element: the conjugacy-class sum makes the character-to-degree ratio an algebraic integer, and the Kronecker theorem on algebraic integers in the unit disk makes a nonzero ratio a root of unity, forcing a scalar in a faithful group representation. Column character orthogonality then makes $1/p$ an algebraic integer, impossible. This is the character-theoretic ingredient in Burnside's theorem.

If an abelian subgroup $A$ has index $p^a$, the centralizer of any $1\ne g\in A$ contains $A$, so the class size of $g$ divides $p^a$. The prime-power conjugacy-class obstruction to simplicity rules this out. If $A=1$, the group is a p-group and has nontrivial center of a group, also ruling out nonabelian simplicity.

The order $|G|$ of a finite group is its number of elements.

A nontrivial group has more than one element.

An identity element $e$ satisfies $eg=ge=g$ for every element $g$ of a group.

The inverse $g^{-1}$ of a group element $g$ satisfies $g{g}^{-1}=g^{-1}g=e$.

A group operation is the associative binary operation of a group. It has an identity element, and every element has an inverse element.

An operation is associative when $(xy)z=x(yz)$ whenever the products are defined.

The group commutator measures the failure of two elements to commute. Under the convention $[x,y]=x^{-1}{y}^{-1}xy$, it is the identity exactly when $xy=yx$.

The Hall-Witt identity is a three-variable identity among iterated group commutators. Passing to the leading terms of a filtered group turns it into the Jacobi identity for the associated graded Lie algebra.

The Hall-Petrescu formula expands $(xy{)}^{p^n}$ as $x^{p^n}{y}^{p^n}$ times ordered powers of higher commutators. In a p-valued group, the correction terms have controlled higher valuation and make the p-power map linear on the associated graded group.

Left and right multiplication by a fixed group element are bijections, called left and right translations.

A cyclic group is generated by one element. A cyclic group of order $n$ has exactly $d$ solutions of $x^d=1$ for every positive integer divisor $d$ of $n$.

The cyclic subgroup generated by an element $g$ is $⟨g⟩=\{g^n:n\in \mathbb{Z}\}$. Its size is the order of $g$.

In a finite group, a cyclic subgroup of order $m$ has exactly $\phi (m)$ generators, where $\phi$ is the Euler totient function. Each element generates exactly one cyclic subgroup, so dividing the number of elements of order $m$ by $\phi (m)$ counts these subgroups. Include the identity subgroup using $m=1$ and $\phi (1)=1$. In a symmetric group, element counts come from cycle type and order of a group element is the least common multiple of cycle lengths.

A finite cyclic group of order $n$ is isomorphic to $C_n=\mathbb{Z}/n\mathbb{Z}$.

For any group $G$, evaluation at a chosen generator of a group $x$ of $C_n$ gives a bijection between group homomorphisms $C_n\to G$ and elements satisfying $g^n=1$. A group homomorphism must send $x^j$ to $g^j$. Conversely, this formula is well-defined because exponents differing by a multiple of $n$ give equal powers, and it respects multiplication. The displayed correspondence is a bijection of sets; it is not claimed to be a group isomorphism for a nonabelian target. In a symmetric group, the order of a group element is the least common multiple of its disjoint cycle lengths, making this correspondence easy to enumerate.

An infinite cyclic group is generated by an element of infinite order and is isomorphic to the additive group of integers.

Every finite subgroup of ${\mathbb{C}}^{\times }$ is cyclic. If its exponent is $n$, it is a subgroup of the cyclic group of $n$th roots of unity.

A generator of a cyclic group $G$ is an element $g$ such that every element of $G$ is an integer power of $g$.

A subset $S$ of a group $G$ is a generating set when every element of $G$ is a finite product of elements of $S$ and their inverses, equivalently when the subgroup $⟨S⟩$ generated by $S$ equals $G$.

For a proper subgroup $H$ of any group $G$, its complement is a generating set of a group. Choose $x\notin H$. For every $h\in H$, also $xh\notin H$, and $h=x^{-1}(xh)$ lies in the subgroup generated by the complement. That subgroup consequently contains both $H$ and its complement. No finiteness assumption is needed.

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

If $G$ has $d$ generators, there are at most $(n!{)}^d$ group homomorphisms $G\to S_n$. Each subgroup of index $n$ is a point stabilizer in a transitive coset group action, so there are at most $n(n!{)}^d$ such subgroups. Consequently finitely many subgroups have index at most any fixed bound. Intersecting them gives a finite-index characteristic subgroup, useful in proving residual finiteness of semidirect products.

The rank $d(G)$ of a finitely generated group is the minimum cardinality of a generating set of a group of $G$.

An abelian group is a group whose operation is commutative: $gh=hg$ for every $g,h$ in the group.

A class of abelian groups closed under isomorphisms, subgroups, quotients and extensions. For the homotopical Serre-class theorems one also uses tensor/Tor closure and homology closure for Eilenberg–MacLane spaces. A homomorphism is an isomorphism modulo the class when its kernel and cokernel lie in the class.

The class of all finitely generated abelian groups has the closure properties needed for the Hurewicz theorem modulo a Serre class. In a simply connected space, degreewise membership of homotopy groups and integral homology groups in this class is equivalent.

An abelian group in which no nonzero element has finite order. Free abelian groups are torsion-free, but torsion-free groups need not be free, as the additive rationals show. This algebraic meaning differs from the torsion-free connection in differential geometry.

For a torsion-free abelian group $G$, its rational divisible hull is $G{\otimes }_{\mathbb{Z}}\mathbb{Q}$. It consists of fractions $g/n$, with $n>0$ and $g/n=h/m$ exactly when $mg=nh$. The natural map $G\to G\otimes \mathbb{Q}$ is injective. Every homomorphism from $G$ into a torsion-free divisible Abelian group extends uniquely through this hull.

An abelian subgroup is a subgroup that is an abelian group under the restricted operation.

A finitely generated abelian group has a finite generating set. Equivalently, it is the cokernel of a homomorphism between finitely generated free abelian groups.

The rank of a finitely generated abelian group $G$ is the number $r$ of infinite cyclic factors in its invariant-factor decomposition, equivalently ${\dim }_{\mathbb{Q}}(G{\otimes }_{\mathbb{Z}}\mathbb{Q})$.

Every finitely generated abelian group is isomorphic toThe rank $r$ and invariant factors $d_i>1$ are unique.

For a prime $p$, the Prüfer group $C_{p^{\infty }}$ is the union of an ascending chain of cyclic groups of orders $p,p^2,p^3,\dots$. As a $\mathbb{Z}$-module it is Artinian but not Noetherian.

An additive group is an abelian group whose operation is written as addition, with identity $0$ and inverse of $x$ written $-x$.

A nonzero finitely generated subgroup of the additive rational numbers is an infinite cyclic group. Put the generators over a common positive denominator. Their integer numerators generate the ideal given by their greatest common divisor, and Bezout identity expresses that divisor as an integer combination of them. Dividing back by the denominator gives a single generator.

An element $a$ of an abelian group is infinitely divisible when, for every positive integer $n$, there is an element $b$ with $nb=a$. Compactness can create a nonzero infinitely divisible element in a nonstandard model even when the original group has none.

A finite abelian group is a finite group whose operation is commutative.

An elementary abelian $p$-group is a finite abelian group in which every nonidentity element has order $p$. It is equivalently a finite-dimensional vector space over ${\mathbb{F}}_p$ under addition.

Pontryagin duality associates to a locally compact abelian group the group of its continuous characters and identifies the double dual canonically with the original group.

The character group $\hat{G}$ consists of the group homomorphisms $\gamma :G\to {\mathbb{C}}^{\times }$. Pointwise multiplication makes it a finite abelian group isomorphic to $G$.

If $H$ is a subgroup of a finite abelian group, $a\notin H$, and $k$ is the least positive integer with $ka\in H$, then each character of a finite abelian group on $H$ has exactly $k$ extensions to $H+⟨a⟩$. The displayed root choice makes the extension well-defined. Starting with the trivial subgroup and adjoining generators proves that the number of characters equals the group order without using a decomposition into cyclic groups.

A character of a finite abelian group is a homomorphism into the complex unit circle. Equivalently it is a linear character; finite order forces each character value to be a root of unity. These are the elements of the character group of a finite abelian group and form the orthogonal Fourier basis of functions on the group.

For a linear character $\chi$ of a finite group $K$,Indeed, if $\chi (h)\ne 1$, the bijection $k\mapsto hk$ gives ${\sum }_k\chi (k)=\chi (h){\sum }_k\chi (k)$, forcing the sum to vanish. The trivial character gives $|K|$ directly. The same argument applies to the restriction of a linear character of a larger group to a subgroup.

For a subgroup $K$ of a finite abelian group $G$, its annihilator isThese are exactly the linear characters that descend to the quotient group $G/K$. The character group of a finite abelian group has the same size as the group, so $|K^{\perp }|=|G/K|=|G|/|K|$. The annihilator is what abelian hidden-subgroup Fourier sampling measures.

A finite additive group is a finite group whose operation is written as addition. Its identity element is denoted $0$, the inverse element of $x$ is denoted $-x$, and translation $y\mapsto x+y$ is a bijection.

A subgroup is a subset of a group that is itself a group under the restricted operation.

A Hall subgroup has order coprime to its index. For a set of primes $\pi$, a Hall $\pi$-subgroup has order using only primes of $\pi$ and index using only primes outside $\pi$. Thus its order contains the full prime-power contributions belonging to $\pi$. A Sylow subgroup is the single-prime instance. Hall subgroup existence in soluble groups generalizes Sylow existence to every prime set.

Every finite soluble group has a Hall subgroup for each prime set. Induct on the group order using an elementary abelian minimal normal subgroup. In the coprime hard case, lift a minimal normal subgroup of the quotient, choose its Sylow subgroup, and apply the Frattini argument. A proper normalizer reduces the order; a normal Sylow subgroup allows induction in its quotient. This proves existence without assuming an independent complement theorem.

The union of two subgroups is a subgroup exactly when one contains the other. Otherwise choose $h\in H\setminus K$ and $k\in K\setminus H$. Their product cannot lie in $H$, since $h^{-1}(hk)=k$ would then lie in $H$, and cannot lie in $K$, since $(hk)k^{-1}=h$ would then lie in $K$. The union therefore fails closure. Unlike the union, the subgroup generated by both subgroups always exists.

The index $[G:H]$ is the number of left cosets of $H$ in $G$.

Every subgroup $H\le G$ of index two is normal: the two left cosets and the two right cosets are both $H$ and its complement, so they agree.

A subgroup $H\le G$ has finite index when the coset space $G/H$ is finite. Intersecting a finite-index subgroup with a subgroup again gives a finite-index subgroup of the latter.