Semi-direct product

In group theory , a branch of mathematics , the semi-direct product (also semi-direct product or entangled product ) represents a special method with which a new group can be constructed from two given groups . This construction generalizes the concept of the direct product of groups and is itself a special case of the concept of group expansion of two groups.

Conversely, if a group with two subgroups is given, the properties of the latter can be used to determine whether it is their semi-direct product.

External semi-direct product

definition

Given are two groups and , as well as a homomorphism of the group into the group of automorphisms of  ${\displaystyle N}$ ${\displaystyle H}$ ${\displaystyle \theta \;\colon H\to \operatorname {Aut} (N)}$ ${\displaystyle H}$ ${\displaystyle N.}$

The Cartesian product of the sets and is the set of all pairs with and It forms with the connection of the pairs ${\displaystyle G=N\times H}$ ${\displaystyle N}$ ${\displaystyle H}$ ${\displaystyle (n,h)}$ ${\displaystyle n\in N}$ ${\displaystyle h\in H.}$ ${\displaystyle \diamond }$

${\displaystyle (n_{1},h_{1})\diamond (n_{2},h_{2}):=(n_{1}\cdot \theta (h_{1})(n_{2}),h_{1}\cdot h_{2})}$           (A)

a group.

proof

The replacement rule ${\displaystyle (n_{1},h_{1})\diamond (n_{3},h_{3})\to (n_{1}\cdot \theta (h_{1})(n_{3}),h_{1}\cdot h_{3})}$ creates the right component of the first operand in the result in the right component and the left component of the second operand in the left. Indeed, the set equipped with this connection satisfies the group axioms. With the inverse is found, because ${\displaystyle N\!\times \!H}$ ${\displaystyle (n^{\prime },h^{\prime }):={\bigl (}\theta (h^{-1})(n^{-1}),h^{-1}{\bigr )}}$ ${\displaystyle {\begin{array}{llrlll}&(n,&h)\diamond &&(n^{\prime },&h^{\prime })\\=&(n,&h)\diamond &{\bigl (}\theta (h^{-1})&(n^{-1}),&h^{-1}{\bigr )}\\=&{\bigl (}n\cdot &\theta (h){\bigl [}&\theta (h^{-1})&(n^{-1}){\bigr ]},h\cdot &h^{-1}{\bigr )}\\=&{\bigl (}n\cdot &{\bigl [}\theta (h)\circ &\theta (h^{-1}){\bigr ]}&(n^{-1}),&h\cdot h^{-1}{\bigr )}\\=&(n\cdot &\theta (1_{H})&&(n^{-1}),&1_{H})\\=&(n\cdot &\operatorname {id} _{\operatorname {Aut} (N)}&&(n^{-1}),&1_{H})\\=&(n\cdot &&&n^{-1},&1_{H})\\=&(1_{N},&&&&1_{H})\\\end{array}}}$ The associative law results as follows: ${\displaystyle {\begin{array}{llrlrlr}&((n_{1},&h_{1})\diamond &(n_{2},&h_{2}))\diamond &(n_{3},&h_{3})\\=&(n_{1}\cdot &\theta (h_{1})&(n_{2}),h_{1}\cdot &h_{2})\diamond &(n_{3},&h_{3})\\=&(n_{1}\cdot &\theta (h_{1})&(n_{2})\cdot &\theta (h_{1}\cdot h_{2})&(n_{3}),h_{1}\cdot h_{2}\cdot &h_{3})\\=&(n_{1}\cdot &\theta (h_{1})&(n_{2})\cdot &\theta (h_{1})\circ \theta (h_{2})&(n_{3}),h_{1}\cdot &h_{2}\cdot h_{3})\\=&(n_{1}\cdot &\theta (h_{1})&(n_{2})\cdot &\theta (h_{1}){\bigl [}\theta (h_{2})&(n_{3}){\bigr ]},h_{1}\cdot &h_{2}\cdot h_{3})\\=&(n_{1}\cdot &\theta (h_{1})&{\bigl [}n_{2}\cdot &\theta (h_{2})&(n_{3}){\bigr ]},h_{1}\cdot &h_{2}\cdot h_{3})\\=&(n_{1},&h_{1})\diamond &(n_{2}\cdot &\theta (h_{2})&(n_{3}),h_{2}\cdot &h_{3})\\=&(n_{1},&h_{1})\diamond &((n_{2},&h_{2})\diamond &(n_{3},&h_{3}))\end{array}}}$

This group is called the (external) semi-direct product of and (by means of ) and is noted as, since the (mediating) homomorphism has a significant influence on the structure of this group. For example, you get the direct product if you choose trivial, i.e. for everyone ${\displaystyle N}$ ${\displaystyle H}$ ${\displaystyle \theta }$ ${\displaystyle N\rtimes _{\theta }H}$ ${\displaystyle \theta }$ ${\displaystyle N\times H,}$ ${\displaystyle \theta }$ ${\displaystyle \theta (h):=\operatorname {id} _{N}\in \operatorname {Aut} (N)}$ ${\displaystyle h\in H.}$

In contrast to the direct product, in this definition the two constituent factors play different roles in the structure of the product. Through the group operates on not the other way around. More precisely: The rule (A) makes the factor the normal divisor with one . If there are different homomorphisms, then if the factors are the same, the semi-direct products are usually different (i.e. not isomorphic). ${\displaystyle \theta }$ ${\displaystyle H}$ ${\displaystyle N,}$ ${\displaystyle \theta \;\colon H\to \operatorname {Aut} (N)}$ ${\displaystyle N}$ ${\displaystyle \theta ,}$

While the direct product does not result in the same but isomorphic structure when the factors are swapped, the group operation from to is missing in the semi-direct product. For similar reasons, an extension to more than two factors makes little sense and is not common in the literature. Pointed, if formulated imprecisely: The semi-direct product is associative, but not commutative. ${\displaystyle N}$ ${\displaystyle H.}$

properties

1. The direct product , which can be constructed into any groups and , is a semi-direct product with the trivial ${\displaystyle N\times H}$ ${\displaystyle N}$ ${\displaystyle H}$ ${\displaystyle \theta .}$
2. If the outer semidirect product has been formed from any two groups and and one , then the group contains with a normal subgroup that is too isomorphic and with a subgroup that is too isomorphic and can be understood as the inner semidirect product of and . ${\displaystyle N}$ ${\displaystyle H}$ ${\displaystyle \theta \;\colon H\to \operatorname {Aut} (N)}$ ${\displaystyle G:=N\rtimes _{\theta }H}$ ${\displaystyle G}$ ${\displaystyle N^{\prime }:=N\times \{1_{H}\}}$ ${\displaystyle N}$ ${\displaystyle H^{\prime }:=\{1_{N}\}\times H}$ ${\displaystyle H}$ ${\displaystyle N^{\prime }}$ ${\displaystyle H^{\prime }}$
3. The group is Abelian if and only if and are Abelian and is trivial. ${\displaystyle N\rtimes _{\theta }H}$ ${\displaystyle N}$ ${\displaystyle H}$ ${\displaystyle \theta }$

Inner semi-direct product

Given a group , a normal subgroup and a subgroup then the following conditions are equivalent: ${\displaystyle G}$ ${\displaystyle N\vartriangleleft G}$ ${\displaystyle H<G,}$

• ${\displaystyle G}$is the complex product , and the subgroups have trivial average ${\displaystyle G=NH}$ ${\displaystyle N\cap H=\{1_{G}\}.}$
• For each there are unique and with ${\displaystyle g\in G}$ ${\displaystyle n\in N}$ ${\displaystyle h\in H}$ ${\displaystyle g=nh.}$
• For each there are unique and with ${\displaystyle g\in G}$ ${\displaystyle n\in N}$ ${\displaystyle h\in H}$ ${\displaystyle g=hn.}$
• There is a homomorphism that is fixed element by element and is its core . ${\displaystyle G\to H}$ ${\displaystyle H}$ ${\displaystyle N}$
• The sequential execution of the embedding and the canonical mapping is an isomorphism ${\displaystyle v\circ r}$ ${\displaystyle r\;\colon H\to G}$ ${\displaystyle v\;\colon G\to G/N}$ ${\displaystyle H\cong G/N.}$

definition

If one of these conditions is met, then the (internal) semi-direct product of and in characters ${\displaystyle G}$ ${\displaystyle N}$ ${\displaystyle H,}$

${\displaystyle N\!\rtimes \!H.}$

The components and play different roles and are generally not interchangeable. The normal divisor is always on the open side of the sign, usually it is noted first. ${\displaystyle N}$ ${\displaystyle H}$ ${\displaystyle \rtimes ,}$

Disintegrating short exact sequence (splitting lemma)

The last two of the above conditions are other formulations of the decay lemma:

• A group is then exactly isomorphic to the semi-direct product of two groups and if there is a short exact sequence are ${\displaystyle G}$ ${\displaystyle N}$ ${\displaystyle H}$

${\displaystyle 1\longrightarrow N\,{\xrightarrow {\ u\ }}\,G\,{\xrightarrow {\ v\ }}\,H\longrightarrow 1}$

as well as a homomorphism so that the identity is on . It is said: the exact sequence disintegrates or disintegrates in the short exact sequence or disintegrates over ${\displaystyle r\;\colon H\to G}$ ${\displaystyle v\circ r=\operatorname {id} _{H}}$ ${\displaystyle H}$ ${\displaystyle G}$ ${\displaystyle G}$ ${\displaystyle N.}$

The homomorphism conveying the semi-direct product is ${\displaystyle N\!\rtimes _{\theta }\!H}$ ${\displaystyle \theta \;\colon H\to \operatorname {Aut} (N)}$

${\displaystyle \theta (h)(n)=u^{-1}\left(r(h)\cdot u(n)\cdot r{\bigl (}h^{-1}{\bigr )}\right).}$

Because of the normal part property of is for all so that is always defined. ${\displaystyle u(N)}$ ${\displaystyle n^{\prime }:=g\cdot u(n)\cdot g^{-1}\in u(N)}$ ${\displaystyle g\in G,}$ ${\displaystyle u^{-1}(n^{\prime })}$

The lemma is a criterion for semi-directness both in the internal and in the external case in which and are not subgroups. ${\displaystyle N}$ ${\displaystyle H}$

Examples

• In the list of small groups , the non-commutative group of order 16 is the semi-direct product without specifying a mediating homomorphism . Now the automorphism group consists of 2 elements, which correspond to the prime remainder classes in . The trivial with mediates as a semidirect product the commutative group The non-commutative semidirect product is mediated by. The following formulas then exist, whereby all information in d. H. modulo 4, to be understood as follows: ${\displaystyle C_{4}\rtimes C_{4}}$ ${\displaystyle \theta }$ ${\displaystyle \operatorname {Aut} (C_{4})={\bigl \{}a\mapsto \alpha a\,{\big |}\,\alpha \in \{1,3\}{\bigr \}}}$ ${\displaystyle C_{4}}$ ${\displaystyle \theta (a)=1^{a}}$ ${\displaystyle a\in \{0,1,2,3\}}$ ${\displaystyle C_{4}\times C_{4}.}$ ${\displaystyle \theta (a)=3^{a}}$ ${\displaystyle \mathbb {Z} /4\mathbb {Z} ,}$

${\displaystyle (a,b)\diamond (c,d)=(a+3^{b}c,b+d),\qquad a,b,c,d\in \{0,1,2,3\}}$

${\displaystyle (0,0)}$   is the neutral element.

${\displaystyle (a,b)^{-1}=(-3^{b}a,-b),\qquad a,b\in \{0,1,2,3\}}$.

In particular, is how you can tell that the group is not commutative. ${\displaystyle (a,1)\diamond (b,1)=(a+3b,2)}$

• There are 4 (non-isomorphic) groups that are semi-direct products of the cyclic groups and . These semidirect products correspond to the 4 automorphisms of the residue class ring , which in turn correspond to the prime residue classes . ${\displaystyle C_{8}=\mathbb {Z} /8\mathbb {Z} }$ ${\displaystyle C_{2}=\mathbb {Z} /2\mathbb {Z} }$ ${\displaystyle \mathbb {Z} /8\mathbb {Z} }$ ${\displaystyle 1,3,5,7\in (\mathbb {Z} /8\mathbb {Z} )^{\times }}$

1. The direct product ${\displaystyle C_{8}\times C_{2}}$ ${\displaystyle (\alpha =1)}$
2. The quasi-dihedral group of order 16 ${\displaystyle (\alpha =3)}$
3. The non-Hamiltonian, nonabelian group of order 16 (English Iwasawa group ) ${\displaystyle (\alpha =5)}$
4. The dihedral group of order 16 ${\displaystyle (\alpha =7)}$

• The unit group of the Hurwitz quaternions is the semi-direct product of the non-commutative quaternion group and the cyclic group with ${\displaystyle Q_{24}:=\left\{\xi \in H\mid \|\xi \|=1\right\}}$ ${\displaystyle H}$ ${\displaystyle {\mathsf {Q}}_{8}\rtimes Q_{3}}$ ${\displaystyle {\mathsf {Q}}_{8}:=\left\{\pm 1,\pm \mathrm {i} ,\pm \mathrm {j} ,\pm \mathrm {k} \right\}}$ ${\displaystyle Q_{3}:=\{1,\varepsilon ^{2},\varepsilon ^{4}\}}$ ${\displaystyle \varepsilon :={\tfrac {1}{2}}(1+\mathrm {i} +\mathrm {j} +\mathrm {k} ).}$

• The group of the automorphisms of a complex or real simple Lie algebra is the semi direct product of the group of inner automorphisms with the group of "outer automorphisms" , that is, decomposes the following short exact sequence: . ${\displaystyle \operatorname {Aut} ({\mathfrak {g}})}$ ${\displaystyle {\mathfrak {g}}}$ ${\displaystyle \operatorname {Inn} ({\mathfrak {g}})=\operatorname {Aut} ({\mathfrak {g}})_{0}}$ ${\displaystyle \operatorname {Out} ({\mathfrak {g}})=\operatorname {Aut} ({\mathfrak {g}})/\operatorname {Aut} ({\mathfrak {g}})_{0}}$ ${\displaystyle 1\rightarrow \operatorname {Aut} ({\mathfrak {g}})_{0}\rightarrow \operatorname {Aut} ({\mathfrak {g}})\rightarrow \operatorname {Aut} ({\mathfrak {g}})/\operatorname {Aut} ({\mathfrak {g}})_{0}\rightarrow 1}$

Finite group theory

• The dihedral group , i.e. the symmetry group of a flat, regular corner, is isomorphic to the semi-direct product of the cyclic rotational symmetry group (which can be described by a cyclic exchange of the corners of the polygon) with a two-element cyclic group . The element operates through it ${\displaystyle D_{n}}$ ${\displaystyle n}$ ${\displaystyle N\cong C_{n}}$ ${\displaystyle H=\langle \sigma \rangle \cong C_{2}}$ ${\displaystyle \sigma }$

${\displaystyle \theta (\sigma )\colon \quad N\to N;\quad g\mapsto g^{-1}}$

on , d. H. the conjugation with σ corresponds to the formation of the inverse in . The element can be understood as a reflection of the polygon on one of its axes of symmetry. ${\displaystyle N}$ ${\displaystyle N}$ ${\displaystyle \sigma }$

• For the symmetric group is isomorphic to a semi-direct product of its normal divisor (the alternating group ) and a two-element cyclic group . The element operates on by swapping the numbers and in the permutation representation of ( ). Construed as an inner semi-direct product: For is the symmetric group a semi-direct product of their normal subgroup with her by any transposition generated subgroup . ${\displaystyle n>1}$ ${\displaystyle S_{n}}$ ${\displaystyle N=A_{n}}$ ${\displaystyle H=\langle \tau _{(jk)}\rangle \cong C_{2}}$ ${\displaystyle \tau =\tau _{(jk)}}$ ${\displaystyle N}$ ${\displaystyle \alpha \in N=A_{n}}$ ${\displaystyle j}$ ${\displaystyle k}$ ${\displaystyle 1\leq j<k\leq n}$ ${\displaystyle n>1}$ ${\displaystyle S_{n}}$ ${\displaystyle A_{n}}$ ${\displaystyle \tau \in S_{n}}$ ${\displaystyle \langle \tau \rangle }$
• The Schur-Zassenhaus theorem is a criterion as to when you can write a finite group as a semi-direct product.

The holomorph of a group

If the homomorphism is used specifically as a mediating one, the semidirect product obtained is the holomorph of ${\displaystyle \theta :=\operatorname {id} _{\operatorname {Aut} (G)}:\operatorname {Aut} (G)\rightarrow \operatorname {Aut} (G)}$ ${\displaystyle G\rtimes _{\theta }\operatorname {Aut} (G)}$ ${\displaystyle G.}$

Application examples in transformation groups

Important examples of semi-direct products are

Euclidean group

One example is the Euclidean group . Every orthogonal matrix describes an automorphism in the space of translations through ${\displaystyle \operatorname {E} (n)=\mathbb {R} ^{n}\rtimes \operatorname {O} (n)}$ ${\displaystyle R\in \operatorname {O} (n)}$ ${\displaystyle T\in \mathbb {R} ^{n}}$

${\displaystyle {\begin{aligned}\theta (R):\;&\mathbb {R} ^{n}\to &\mathbb {R} ^{n}\\&T\mapsto &R\cdot T\,.\end{aligned}}}$

A movement operates on points by and it is ${\displaystyle (T,R)\in \operatorname {E} (n)}$ ${\displaystyle p\in \mathbb {R} ^{n}}$ ${\displaystyle (T,R)[p]:=T+R\cdot p}$

${\displaystyle (T_{1},R_{1})[(T_{2},R_{2})[p]]=T_{1}+R_{1}(T_{2}+R_{2}p)=(T_{1}+R_{1}\cdot T_{2},R_{1}\cdot R_{2})[p]}$.

The following applies to products in : ${\displaystyle \operatorname {E} (n)}$

${\displaystyle (T_{1},R_{1})\diamond (T_{2},R_{2})=(T_{1}+\theta (R_{1})[T_{2}],R_{1}\cdot R_{2})}$ .

This product is not Abelian because it applies to and : ${\displaystyle R\neq \mathbf {1} }$ ${\displaystyle T\neq \mathbf {0} }$

${\displaystyle {\begin{aligned}&(T,\mathbf {1} )\diamond (\mathbf {0} ,R)=&(T,R)\\\neq \;&(\mathbf {0} ,R)\diamond (T,\mathbf {1} )=&(RT,R)\end{aligned}}}$

Poincaré group

The Poincaré group , which is the semi-direct product of the group of translations and the group of Lorentz transformations . The element from denotes a displacement with the vector . The homomorphism is then given by for every Lorentz transformation and every vector . The Poincaré group is particularly important for the special theory of relativity , where it appears as an invariance group . ${\displaystyle N=\mathbb {R} ^{3+1}}$ ${\displaystyle H=O(3,1)}$ ${\displaystyle T_{a}}$ ${\displaystyle N}$ ${\displaystyle a\in \mathbb {R} ^{3+1}}$ ${\displaystyle \theta }$ ${\displaystyle \theta (L)(T_{a})=T_{La}}$ ${\displaystyle L}$ ${\displaystyle a}$

See also

• Affine group
• Semi-direct sum

Web links

• Rudolf Scharlau Algebra I 2.7 Semi-Direct Products

literature

• Thomas W. Hungerford: Algebra . 5. Springer-Verlag, 1989, ISBN 0-387-90518-9

Individual evidence

1. ↑ JLT 20035. Retrieved December 13, 2019 .