Semidirect product

Inmathematics,specifically ingroup theory,the concept of asemidirect productis a generalization of adirect product.There are two closely related concepts of semidirect product:

• aninnersemidirect product is a particular way in which agroupcan be made up of twosubgroups,one of which is anormal subgroup.
• anoutersemidirect product is a way to construct a new group from two given groups by using theCartesian productas a set and a particular multiplication operation.

As with direct products, there is a natural equivalence between inner and outer semidirect products, and both are commonly referred to simply assemidirect products.

Forfinite groups,theSchur–Zassenhaus theoremprovides a sufficient condition for the existence of a decomposition as a semidirect product (also known assplitting extension).

Inner semidirect product definitions

Given a groupGwithidentity elemente,asubgroupH,and anormal subgroupN◁G,the following statements are equivalent:

• Gis theproduct of subgroups,G=NH,and these subgroups have trivial intersection:N∩H= {e}.
• For everyg∈G,there are uniquen∈Nandh∈Hsuch thatg=nh.
• Thecompositionπ∘iof the natural embeddingi:H→Gwith the natural projectionπ:G→G/Nis anisomorphismbetweenHand thequotient groupG/N.
• There exists ahomomorphismG→Hthat is theidentityonHand whosekernelisN.In other words, there is a splitexact sequence

$1\to N\to G\to H\to 1$

of groups (which is also known as group extension of$H$by$N$).

If any of these statements holds (and hence all of them hold, by their equivalence), we sayGis thesemidirect productofNandH,written

$G=N\rtimes H$or$G=H\ltimes N,$

or thatGsplitsoverN;one also says thatGis asemidirectproduct ofHacting onN,or even a semidirect product ofHandN.To avoid ambiguity, it is advisable to specify which is the normal subgroup.

If$G=N\rtimes H$,then there is a group homomorphism$\varphi \colon H\rightarrow \mathrm {Aut} (N)$given by$\varphi _{h}(n)=hnh^{-1}$,and for$g=nh,g'=n'h'$,we have$gg'=nhn'h'=nhn'h^{-1}hh'=n\varphi _{h}(n')hh'=n^{*}h^{*}$.

Inner and outer semidirect products

Let us first consider the inner semidirect product. In this case, for a group$G$,consider a normal subgroupNand another subgroupH(not necessarily normal). Assume that the conditions on the list above hold. Let$\operatorname {Aut} (N)$denote the group of allautomorphismsofN,which is a group under composition. Construct a group homomorphism$\varphi \colon H\to \operatorname {Aut} (N)$defined by conjugation,

$\varphi _{h}(n)=hnh^{-1}$,for allhinHandninN.

In this way we can construct a group$G'=(N,H)$with group operation defined as

$(n_{1},h_{1})\cdot (n_{2},h_{2})=(n_{1}\varphi _{h_{1}}(n_{2}),\,h_{1}h_{2})$forn₁,n₂inNandh₁,h₂inH.

The subgroupsNandHdetermineGup toisomorphism, as we will show later. In this way, we can construct the groupGfrom its subgroups. This kind of construction is called aninner semidirect product(also known as internal semidirect product).

Let us now consider the outer semidirect product. Given any two groupsNandHand a group homomorphismφ:H→ Aut(N),we can construct a new groupN⋊_φH,called theouter semidirect productofNandHwith respect toφ,defined as follows:

• The underlying set is theCartesian productN×H.
• The group operation$\bullet$is determined by the homomorphismφ:

  ${\begin{aligned}\bullet:(N\rtimes _{\varphi }H)\times (N\rtimes _{\varphi }H)&\to N\rtimes _{\varphi }H\\(n_{1},h_{1})\bullet (n_{2},h_{2})&=(n_{1}\varphi _{h_{1}}(n_{2}),\,h_{1}h_{2})\end{aligned}}$

  forn₁,n₂inNandh₁,h₂inH.

This defines a group in which the identity element is(e_N,e_H)and the inverse of the element(n,h)is(φ_h⁻¹(n⁻¹),h⁻¹).Pairs(n,e_H)form a normal subgroup isomorphic toN,while pairs(e_N,h)form a subgroup isomorphic toH.The full group is a semidirect product of those two subgroups in the sense given earlier.

Conversely, suppose that we are given a groupGwith a normal subgroupNand a subgroupH,such that every elementgofGmay be written uniquely in the formg = nhwherenlies inNandhlies inH.Letφ:H→ Aut(N)be the homomorphism (writtenφ(h) =φ_h) given by

$\varphi _{h}(n)=hnh^{-1}$

for alln∈N,h∈H.

ThenGis isomorphic to the semidirect productN⋊_φH.The isomorphismλ:G→N⋊_φHis well defined byλ(a) =λ(nh) = (n, h)due to the uniqueness of the decompositiona=nh.

InG,we have

$(n_{1}h_{1})(n_{2}h_{2})=n_{1}h_{1}n_{2}(h_{1}^{-1}h_{1})h_{2}=(n_{1}\varphi _{h_{1}}(n_{2}))(h_{1}h_{2})$

Thus, fora=n₁h₁andb=n₂h₂we obtain

${\begin{aligned}\lambda (ab)&=\lambda (n_{1}h_{1}n_{2}h_{2})=\lambda (n_{1}\varphi _{h_{1}}(n_{2})h_{1}h_{2})=(n_{1}\varphi _{h_{1}}(n_{2}),h_{1}h_{2})=(n_{1},h_{1})\bullet (n_{2},h_{2})\\[5pt]&=\lambda (n_{1}h_{1})\bullet \lambda (n_{2}h_{2})=\lambda (a)\bullet \lambda (b),\end{aligned}}$

whichprovesthatλis a homomorphism. Sinceλis obviously an epimorphism and monomorphism, then it is indeed an isomorphism. This also explains the definition of the multiplication rule inN⋊_φH.

The direct product is a special case of the semidirect product. To see this, letφbe the trivial homomorphism (i.e., sending every element ofHto the identity automorphism ofN) thenN⋊_φHis the direct productN×H.

A version of thesplitting lemmafor groups states that a groupGis isomorphic to a semidirect product of the two groupsNandHif and only ifthere exists ashort exact sequence

$1\longrightarrow N\,{\overset {\beta }{\longrightarrow }}\,G\,{\overset {\ Alpha }{\longrightarrow }}\,H\longrightarrow 1$

and a group homomorphismγ:H→Gsuch thatα ∘ γ= id_H,the identity map onH.In this case,φ:H→ Aut(N)is given byφ(h) =φ_h,where

$\varphi _{h}(n)=\beta ^{-1}(\gamma (h)\beta (n)\gamma (h^{-1})).$

Examples

Dihedral group

Thedihedral groupD_2nwith2nelements is isomorphic to a semidirect product of thecyclic groupsC_nandC₂.Here, the non-identity element ofC₂acts onC_nby inverting elements; this is an automorphism sinceC_nisabelian.Thepresentationfor this group is:

$\langle a,\;b\mid a^{2}=e,\;b^{n}=e,\;aba^{-1}=b^{-1}\rangle.$

Cyclic groups

More generally, a semidirect product of any two cyclic groupsC_mwith generatoraandC_nwith generatorbis given by one extra relation,aba⁻¹=b^k,withkandncoprime,and$k^{m}\equiv 1{\pmod {n}}$;that is, the presentation:

$\langle a,\;b\mid a^{m}=e,\;b^{n}=e,\;aba^{-1}=b^{k}\rangle.$

Ifrandmare coprime,a^ris a generator ofC_manda^rba^−r=b^kr,hence the presentation:

$\langle a,\;b\mid a^{m}=e,\;b^{n}=e,\;aba^{-1}=b^{k^{r}}\rangle$

gives a group isomorphic to the previous one.

Holomorph of a group

One canonical example of a group expressed as a semi-direct product is theholomorphof a group. This is defined as

$\operatorname {Hol} (G)=G\rtimes \operatorname {Aut} (G)$

where${\text{Aut}}(G)$is theautomorphism groupof a group$G$and the structure map$\varphi$comes from the right action of${\text{Aut}}(G)$on$G$.In terms of multiplying elements, this gives the group structure

$(g,\ Alpha )(h,\beta )=(g(\varphi (\ Alpha )\cdot h),\ Alpha \beta ).$

Fundamental group of the Klein bottle

Thefundamental groupof theKlein bottlecan be presented in the form

$\langle a,\;b\mid aba^{-1}=b^{-1}\rangle.$

and is therefore a semidirect product of the group of integers,$\mathbb {Z}$,with$\mathbb {Z}$.The corresponding homomorphismφ:$\mathbb {Z}$→ Aut($\mathbb {Z}$)is given byφ(h)(n) = (−1)^hn.

Upper triangular matrices

The group$\mathbb {T} _{n}$of uppertriangular matriceswith non-zerodeterminantin an arbitrary field, that is with non-zero entries on thediagonal,has a decomposition into the semidirect product $\mathbb {T} _{n}\cong \mathbb {U} _{n}\rtimes \mathbb {D} _{n}$where$\mathbb {U} _{n}$is the subgroup ofmatriceswith only$1$'s on the diagonal, which is called the upperunitriangular matrixgroup, and$\mathbb {D} _{n}$is the subgroup ofdiagonal matrices.
The group action of$\mathbb {D} _{n}$on$\mathbb {U} _{n}$is induced by matrix multiplication. If we set

$A={\begin{bmatrix}x_{1}&0&\cdots &0\\0&x_{2}&\cdots &0\\\vdots &\vdots &&\vdots \\0&0&\cdots &x_{n}\end{bmatrix}}$ and$B={\begin{bmatrix}1&a_{12}&a_{13}&\cdots &a_{1n}\\0&1&a_{23}&\cdots &a_{2n}\\\vdots &\vdots &\vdots &&\vdots \\0&0&0&\cdots &1\end{bmatrix}}$

then theirmatrix productis

$AB={\begin{bmatrix}x_{1}&x_{1}a_{12}&x_{1}a_{13}&\cdots &x_{1}a_{1n}\\0&x_{2}&x_{2}a_{23}&\cdots &x_{2}a_{2n}\\\vdots &\vdots &\vdots &&\vdots \\0&0&0&\cdots &x_{n}\end{bmatrix}}.$

This gives the induced group action$m:\mathbb {D} _{n}\times \mathbb {U} _{n}\to \mathbb {U} _{n}$

$m(A,B)={\begin{bmatrix}1&x_{1}a_{12}&x_{1}a_{13}&\cdots &x_{1}a_{1n}\\0&1&x_{2}a_{23}&\cdots &x_{2}a_{2n}\\\vdots &\vdots &\vdots &&\vdots \\0&0&0&\cdots &1\end{bmatrix}}.$

A matrix in$\mathbb {T} _{n}$can be represented by matrices in$\mathbb {U} _{n}$and$\mathbb {D} _{n}$.Hence$\mathbb {T} _{n}\cong \mathbb {U} _{n}\rtimes \mathbb {D} _{n}$.

Group of isometries on the plane

TheEuclidean groupof all rigid motions (isometries) of the plane (mapsf:$\mathbb {R}$²→$\mathbb {R}$²such that the Euclidean distance betweenxandyequals the distance betweenf(x)andf(y)for allxandyin$\mathbb {R} ^{2}$) is isomorphic to a semidirect product of the abelian group$\mathbb {R} ^{2}$(which describes translations) and the groupO(2)oforthogonal2 × 2matrices (which describes rotations and reflections that keep the origin fixed). Applying a translation and then a rotation or reflection has the same effect as applying the rotation or reflection first and then a translation by the rotated or reflected translation vector (i.e. applying theconjugateof the original translation). This shows that the group of translations is a normal subgroup of the Euclidean group, that the Euclidean group is a semidirect product of the translation group andO(2),and that the corresponding homomorphismφ:O(2) → Aut($\mathbb {R}$²)is given bymatrix multiplication:φ(h)(n) =hn.

Orthogonal group O(n)

Theorthogonal groupO(n)of all orthogonalrealn×nmatrices (intuitively the set of all rotations and reflections ofn-dimensional space that keep the origin fixed) is isomorphic to a semidirect product of the groupSO(n)(consisting of all orthogonal matrices withdeterminant1,intuitively the rotations ofn-dimensional space) andC₂.If we representC₂as the multiplicative group of matrices{I,R},whereRis a reflection ofn-dimensional space that keeps the origin fixed (i.e., an orthogonal matrix with determinant–1representing aninvolution), thenφ:C₂→ Aut(SO(n))is given byφ(H)(N) =HNH⁻¹for allHinC₂andNinSO(n).In the non-trivial case (His not the identity) this means thatφ(H)is conjugation of operations by the reflection (in 3-dimensional space a rotation axis and the direction of rotation are replaced by their "mirror image" ).

Semi-linear transformations

The group ofsemilinear transformationson a vector spaceVover a field$\mathbb {K}$,often denotedΓL(V),is isomorphic to a semidirect product of thelinear groupGL(V)(anormal subgroupofΓL(V)), and theautomorphism groupof$\mathbb {K}$.

Crystallographic groups

Incrystallography,the space group of a crystal splits as the semidirect product of the point group and the translation group if and only if the space group issymmorphic.Non-symmorphic space groups have point groups that are not even contained as subset of the space group, which is responsible for much of the complication in their analysis.

Non-examples

Of course, nosimple groupcan be expressed as a semi-direct product (because they do not have nontrivial normal subgroups), but there are a few common counterexamples of groups containing a non-trivial normal subgroup that nonetheless cannot be expressed as a semi-direct product. Note that although not every group$G$can be expressed as a split extension of$H$by$A$,it turns out that such a group can be embedded into thewreath product$A\wr H$by theuniversal embedding theorem.

Z₄

The cyclic group$\mathbb {Z} _{4}$is not a simple group since it has a subgroup of order 2, namely$\{0,2\}\cong \mathbb {Z} _{2}$is a subgroup and their quotient is$\mathbb {Z} _{2}$,so there's an extension

$0\to \mathbb {Z} _{2}\to \mathbb {Z} _{4}\to \mathbb {Z} _{2}\to 0$

If the extension wassplit,then the group$G$in

$0\to \mathbb {Z} _{2}\to G\to \mathbb {Z} _{2}\to 0$

would be isomorphic to$\mathbb {Z} _{2}\times \mathbb {Z} _{2}$.

Q₈

Thegroup of the eight quaternions$\{\pm 1,\pm i,\pm j,\pm k\}$where$ijk=-1$and$i^{2}=j^{2}=k^{2}=-1$,is another example of a groupwhich has non-trivial normal subgroups yet is still not split. For example, the subgroup generated by$i$is isomorphic to$\mathbb {Z} _{4}$and is normal. It also has a subgroup of order$2$generated by$-1$.This would mean$Q_{8}$would have to be a split extension in the followinghypotheticalexact sequence of groups:

$0\to \mathbb {Z} _{4}\to Q_{8}\to \mathbb {Z} _{2}\to 0$,

but such an exact sequence does not exist. This can be shown by computing the first group cohomology group of$\mathbb {Z} _{2}$with coefficients in$\mathbb {Z} _{4}$,so$H^{1}(\mathbb {Z} _{2},\mathbb {Z} _{4})\cong \mathbb {Z} /2$and noting the two groups in these extensions are$\mathbb {Z} _{2}\times \mathbb {Z} _{4}$and the dihedral group$D_{8}$.But, as neither of these groups is isomorphic with$Q_{8}$,the quaternion group is not split. This non-existence of isomorphisms can be checked by noting the trivial extension is abelian while$Q_{8}$is non-abelian, and noting the only normal subgroups are$\mathbb {Z} _{2}$and$\mathbb {Z} _{4}$,but$Q_{8}$has three subgroups isomorphic to$\mathbb {Z} _{4}$.

Properties

IfGis the semidirect product of the normal subgroupNand the subgroupH,and bothNandHare finite, then theorderofGequals the product of the orders ofNandH.This follows from the fact thatGis of the same order as the outer semidirect product ofNandH,whose underlying set is theCartesian productN×H.

Relation to direct products

SupposeGis a semidirect product of the normal subgroupNand the subgroupH.IfHis also normal inG,or equivalently, if there exists a homomorphismG→Nthat is the identity onNwith kernelH,thenGis thedirect productofNandH.

The direct product of two groupsNandHcan be thought of as the semidirect product ofNandHwith respect toφ(h) = id_Nfor allhinH.

Note that in a direct product, the order of the factors is not important, sinceN×His isomorphic toH×N.This is not the case for semidirect products, as the two factors play different roles.

Furthermore, the result of a (proper) semidirect product by means of a non-trivial homomorphism is never anabelian group,even if the factor groups are abelian.

Non-uniqueness of semidirect products (and further examples)

As opposed to the case with thedirect product,a semidirect product of two groups is not, in general, unique; ifGandG′are two groups that both contain isomorphic copies ofNas a normal subgroup andHas a subgroup, and both are a semidirect product ofNandH,then it doesnotfollow thatGandG′areisomorphicbecause the semidirect product also depends on the choice of an action ofHonN.

For example, there are four non-isomorphic groups of order 16 that are semidirect products ofC₈andC₂;in this case,C₈is necessarily a normal subgroup because it has index 2. One of these four semidirect products is the direct product, while the other three are non-abelian groups:

• the dihedral group of order 16
• thequasidihedral groupof order 16
• theIwasawa groupof order 16

If a given group is a semidirect product, then there is no guarantee that this decomposition is unique. For example, there is a group of order 24 (the only one containing six elements of order 4 and six elements of order 6) that can be expressed as semidirect product in the following ways:(D₈⋉ C₃) ≅ (C₂⋉Q₁₂) ≅ (C₂⋉ D₁₂) ≅ (D₆⋉V).

Existence

Main article:Schur–Zassenhaus theorem

In general, there is no known characterization (i.e., a necessary and sufficient condition) for the existence of semidirect products in groups. However, some sufficient conditions are known, which guarantee existence in certain cases. For finite groups, theSchur–Zassenhaus theoremguarantees existence of a semidirect product when theorderof the normal subgroup iscoprimeto the order of thequotient group.

For example, the Schur–Zassenhaus theorem implies the existence of a semi-direct product among groups of order 6; there are two such products, one of which is a direct product, and the other a dihedral group. In contrast, the Schur–Zassenhaus theorem does not say anything about groups of order 4 or groups of order 8 for instance.

Generalizations

Within group theory, the construction of semidirect products can be pushed much further. TheZappa–Szép productof groups is a generalization that, in its internal version, does not assume that either subgroup is normal.

There is also a construction inring theory,thecrossed product of rings.This is constructed in the natural way from thegroup ringfor a semidirect product of groups. The ring-theoretic approach can be further generalized to thesemidirect sum of Lie algebras.

For geometry, there is also a crossed product forgroup actionson atopological space;unfortunately, it is in general non-commutative even if the group is abelian. In this context, the semidirect product is thespace of orbitsof the group action. The latter approach has been championed byAlain Connesas a substitute for approaches by conventional topological techniques; c.f.noncommutative geometry.

The semidirect product is a special case of theGrothendieck constructionincategory theory.Specifically, an action of$H$on$N$(respecting the group, or even just monoid structure) is the same thing as afunctor

$F:BH\to Cat$

from thegroupoid$BH$associated toH(having a single object *, whose endomorphisms areH) to the category of categories such that the unique object in$BH$is mapped to$BN$.The Grothendieck construction of this functor is equivalent to$B(H\rtimes N)$,the (groupoid associated to) semidirect product.

Groupoids

Another generalization is for groupoids. This occurs in topology because if a groupGacts on a spaceXit also acts on thefundamental groupoidπ₁(X)of the space. The semidirect productπ₁(X) ⋊Gis then relevant to finding the fundamental groupoid of theorbit spaceX/G.For full details see Chapter 11 of the book referenced below, and also some details in semidirect productinncatlab.

Abelian categories

Non-trivial semidirect products donotarise inabelian categories,such as thecategory of modules.In this case, thesplitting lemmashows that every semidirect product is a direct product. Thus the existence of semidirect products reflects a failure of the category to be abelian.

Notation

Usually the semidirect product of a groupHacting on a groupN(in most cases by conjugation as subgroups of a common group) is denoted byN⋊HorH⋉N.However, some sourcesmay use this symbol with the opposite meaning. In case the actionφ:H→ Aut(N)should be made explicit, one also writesN⋊_φH.One way of thinking about theN⋊Hsymbol is as a combination of the symbol for normal subgroup (◁) and the symbol for the product (×).Barry Simon,in his book on group representation theory,employs the unusual notation$N\mathbin {\circledS _{\varphi }} H$for the semidirect product.

Unicodelists four variants:

	Value	MathML	Unicode description
⋉	U+22C9	ltimes	LEFT NORMAL FACTOR SEMIDIRECT PRODUCT
⋊	U+22CA	rtimes	RIGHT NORMAL FACTOR SEMIDIRECT PRODUCT
⋋	U+22CB	lthree	LEFT SEMIDIRECT PRODUCT
⋌	U+22CC	rthree	RIGHT SEMIDIRECT PRODUCT

Here the Unicode description of the rtimes symbol says "right normal factor", in contrast to its usual meaning in mathematical practice.

InLaTeX,the commands \rtimes and \ltimes produce the corresponding characters. With the AMS symbols package loaded, \leftthreetimes produces ⋋ and \rightthreetimes produces ⋌.

See also

• Affine Lie algebra
• Grothendieck construction,a categorical construction that generalizes the semidirect product
• Holomorph
• Lie algebra semidirect sum
• Subdirect product
• Wreath product
• Zappa–Szép product
• Crossed product

References

• Barr, Michael; Wells, Charles (2012),Category theory for computing science,Reprints in Theory and Applications of Categories, vol. 2012, p. 558,Zbl1253.18001
• Brown, R. (2006),Topology and groupoids,Booksurge,ISBN1-4196-2722-8