Semidirect product

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In mathematics, specifically in group theory, the concept of a semidirect product is a generalization of a direct product. There are two closely related concepts of semidirect product: an inner semidirect product is a particular way in which a group can be constructed from two subgroups, one of which is a normal subgroup, while an outer semidirect product is a cartesian product as a set, but with 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 as semidirect products.

Some equivalent definitions of inner semidirect products[edit]

Let G be a group with identity element e, N a normal subgroup of G (i.e., NG) and H a subgroup of G. The following statements are equivalent:

  • G = NH and NH = {e}.
  • Every element of G can be written in a unique way as a product nh, with nN and hH.
  • Every element of G can be written in a unique way as a product hn, with hH and nN.
  • The natural embedding HG, composed with the natural projection GG / N, yields an isomorphism between H and the quotient group G / N.
  • There exists a homomorphism GH which is the identity on H and whose kernel is N.

If one (and therefore all) of these statements hold, we say that G is a semidirect product of N and H, written G = N \rtimes H, or that G splits over N; one also says that G is a semidirect product of H acting on N, or even a semidirect product of H and N. In order to avoid ambiguities, it is advisable to specify which of the two subgroups is normal.

Elementary facts and caveats[edit]

If G is the semidirect product of the normal subgroup N and the subgroup H, and both N and H are finite, then the order of G equals the product of the orders of N and H.

Note that, as opposed to the case with the direct product, a semidirect product of two groups is not, in general, unique; if G and G′ are two groups which both contain isomorphic copies of N as a normal subgroup and H as a subgroup, and both are a semidirect product of N and H, then it does not follow that G and G′ are isomorphic. This remark leads to an extension problem, of describing the possibilities.

Semidirect products and group homomorphisms[edit]

Let G be a semidirect product of the normal subgroup N and the subgroup H. Let Aut(N) denote the group of all automorphisms of N. The map φ : H → Aut(N) defined by φ(h) = φh, where φh(n) = hnh−1 for all h in H and n in N, is a group homomorphism. (Note that hnh−1N since N is normal in G.) Together N, H and φ determine G up to isomorphism, as we show now.

Given any two groups N and H (not necessarily subgroups of a given group) and a group homomorphism \varphi : H → Aut(N), we can construct a new group N\rtimes_{\varphi}H, called the (outer) semidirect product of N and H with respect to \varphi, defined as follows.[1]

  • As a set, N\rtimes_{\varphi}H is the cartesian product N × H.
  • Multiplication of elements in N\rtimes_{\varphi}H is determined by the homomorphism \varphi. The operation is
*\colon (N\rtimes_{\varphi} H)\times(N\rtimes_{\varphi} H)\to N\rtimes_{\varphi} H
defined by
(n_1, h_1)*(n_2, h_2) = (n_1\varphi_{h_1}(n_2), h_1h_2)
for n1, n2 in N and h1, h2 in H.

This defines a group in which the identity element is (eN, eH) and the inverse of the element (n, h) is (\varphih−1(n−1), h−1). Pairs (n,eH) form a normal subgroup isomorphic to N, while pairs (eN, h) form a subgroup isomorphic to H. The full group is a semidirect product of those two subgroups in the sense given earlier.

Conversely, suppose that we are given a group G with a normal subgroup N and a subgroup H, such that every element g of G may be written uniquely in the form g=nh where n lies in N and h lies in H. Let \varphi: H → Aut(N) be the homomorphism given by \varphi(h) = \varphih, where

\varphi_h(n) = hnh^{-1}

for all n in N and h in H. Then G is isomorphic to the semidirect product N\rtimes_{\varphi}H ; the isomorphism sends the product nh to the tuple (n,h). In G, we have

(n_1h_1)(n_2h_2) = n_1 h_1 n_2 h_1^{-1}h_1h_2 = (n_1\varphi_{h_1}(n_2))(h_1h_2)

which shows that the above map is indeed an isomorphism and also explains the definition of the multiplication rule in N\rtimes_{\varphi}H.

A version of the splitting lemma for groups states that a group G is isomorphic to a semidirect product of the two groups N and H if and only if there exists a short exact sequence

 1\longrightarrow N \longrightarrow^{\!\!\!\!\!\!\!\!\!\beta}\ \, G \longrightarrow^{\!\!\!\!\!\!\!\!\!\alpha}\ \,  H \longrightarrow 1

and a group homomorphism γ : HG such that \alpha \circ \gamma = \operatorname{id}_H, the identity map on H. In this case, \varphi : H → Aut(N) is given by \varphi(h) = \varphih, where

\varphi_h(n) = \beta^{-1}(\gamma(h)\beta(n)\gamma(h^{-1})).

If \varphi is the trivial homomorphism, sending every element of H to the identity automorphism of N, then N\rtimes_{\varphi}H is the direct product  N \times H.

Examples[edit]

The dihedral group D2n with 2n elements is isomorphic to a semidirect product of the cyclic groups Cn and C2.[2] Here, the non-identity element of C2 acts on Cn by inverting elements; this is an automorphism since Cn is abelian. The presentation for this group is:

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

More generally, a semidirect product of any two cyclic groups C_m\; with generator a\; and C_n\; with generator b\; is given by a single relation aba^{-1}=b^k\; with k\; and n\; coprime, i.e. the presentation:[2]

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

If r\; and m\; are coprime, a^r\; is a generator of C_m\; and a^rba^{-r}=b^{k^r}\;, 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.

The fundamental group of the Klein bottle can 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 homomorphism \varphi : \mathbb{Z} \to \mathrm{Aut}(\mathbb{Z}) is given by \varphi(h)(n)=(-1)^h n.

The Euclidean group of all rigid motions (isometries) of the plane (maps f : R2R2 such that the Euclidean distance between x and y equals the distance between f(x) and f(y) for all x and y in R2) is isomorphic to a semidirect product of the abelian group R2 (which describes translations) and the group O(2) of orthogonal 2×2 matrices (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 the conjugate of 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 and O(2), and that the corresponding homomorphism \varphi : O(2) \to \mathrm{Aut}(\mathbb{R}^2) is given by matrix multiplication: \varphi(h)(n)=hn.

The orthogonal group O(n) of all orthogonal real n×n matrices (intuitively the set of all rotations and reflections of n-dimensional space which keep the origin fixed) is isomorphic to a semidirect product of the group SO(n) (consisting of all orthogonal matrices with determinant 1, intuitively the rotations of n-dimensional space) and C2. If we represent C2 as the multiplicative group of matrices {I, R}, where R is a reflection of n dimensional space which keeps the origin fixed (i.e. an orthogonal matrix with determinant –1 representing an involution), then φ : C2 → Aut(SO(n)) is given by φ(H)(N) = H N H−1 for all H in C2 and N in SO(n). In the non-trivial case ( H is not the identity) this means that φ(H) is conjugation of operations by the reflection (a rotation axis and the direction of rotation are replaced by their "mirror image").

The group of semilinear transformations on a vector space V over a field \mathbb{K}, often denoted \operatorname{\Gamma L}(V), is isomorphic to a semidirect product of the linear group \operatorname{GL}(V) (a normal subgroup of \operatorname{\Gamma L}(V)), and the automorphism group of \mathbb{K}.

Relation to direct products[edit]

Suppose G is a semidirect product of the normal subgroup N and the subgroup H. If H is also normal in G, or equivalently, if there exists a homomorphism GN which is the identity on N, then G is the direct product of N and H.

The direct product of two groups N and H can be thought of as the semidirect product of N and H with respect to φ(h) = idN for all h in H.

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

Generalizations[edit]

The construction of semidirect products can be pushed much further. The Zappa–Szep product of groups is a generalization which, in its internal version, does not assume that either subgroup is normal. There is also a construction in ring theory, the crossed product of rings. This is seen naturally as soon as one constructs a group ring for a semidirect product of groups. There is also the semidirect sum of Lie algebras. Given a group action on a topological space, there is a corresponding crossed product which will in general be non-commutative even if the group is abelian. This kind of ring (see crossed product for a related construction) can play the role of the space of orbits of the group action, in cases where that space cannot be approached by conventional topological techniques – for example in the work of Alain Connes (cf. noncommutative geometry).

There are also far-reaching generalisations in category theory. They show how to construct fibred categories from indexed categories. This is an abstract form of the outer semidirect product construction.

Groupoids[edit]

Another generalization is for groupoids. This occurs in topology because if a group G acts on a space X it also acts on the fundamental groupoid \pi_1(X) of the space. The semidirect product \pi_1(X) \rtimes G is then relevant to finding the fundamental groupoid of the orbit space X/G. For full details see Chapter 11 of the book referenced below, and also some details in semidirect product[3] in ncatlab.

Abelian categories[edit]

Non-trivial semidirect products do not arise in abelian categories, such as the category of modules. In this case, the splitting lemma shows that every semidirect product is a direct product. Thus the existence of semidirect products reflects a failure of the category to be abelian.

Notation[edit]

Usually the semidirect product of a group H acting on a group N (in most cases by conjugation as subgroups of a common group) is denoted by N\rtimes H or H\ltimes N. However, some sources may use this symbol with the opposite meaning. In case the action \phi :H \rightarrow \operatorname{Aut}(N) should be made explicit, one also writes N\rtimes_{\phi}H. One way of thinking about the N\rtimes H symbol is as a combination of the symbol for normal subgroup (\triangleleft) and the symbol for the product (\times).

Unicode lists four variants:[4]

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.

In LaTeX, the commands \rtimes and \ltimes produce the corresponding characters.

See also[edit]

Notes[edit]

  1. ^ Robinson, Derek John Scott (2003). An Introduction to Abstract Algebra. Walter de Gruyter. pp. 75–76. ISBN 9783110175448. 
  2. ^ a b Mac Lane, Saunders; Birkhoff, Garrett (1999). Algebra (3 ed.). American Mathematical Society. pp. 414–415. ISBN 0-8218-1646-2. 
  3. ^ Ncatlab.org
  4. ^ See unicode.org

References[edit]