Semidirect product

A semidirect product combines a normal subgroup with a subgroup acting on it by automorphisms.

Let an abelian group $V$ carry an action $\rho$ of a group $H$ by automorphisms, and use multiplication $(h,v)(k,w)=(hk,v+\rho (h)w)$. Comparing $(e,v)(h,w)$ and $(h,w)(e,v)$ shows that a translation is central precisely when its vector is fixed by the entire action. If $H$ is itself abelian, the whole semidirect product is abelian exactly when the action is trivial. For $H=(\mathbb{R},+)$, $V={\mathbb{R}}^2$, and determinant-one linear actions fixing the first coordinate vector, $\rho (x)=\left(\begin{array}{cc}1& r(x)\\ 0& 1\end{array}\right)$, where $r$ is an additive function. Without a regularity assumption, it need not be $r(x)=cx$.

If $N=⟨A\mid R⟩$ and $H=⟨B\mid S⟩$ have finite group presentations and $\varphi :H\to \mathrm{Aut}(N)$ is an action, choose a word $w_{b,a}(A)$ representing $\varphi (b)(a)$. Then $N{⋊}_{\varphi }H$ has the finite presentationThe action automorphisms also let these relations rewrite conjugation by inverse generators. Every word becomes an $A$-word followed by a $B$-word. The natural map to the semidirect product is injective because its triviality forces the $B$-word to be trivial using $S$ and then the $A$-word to be trivial using $R$.

If $N$ is an elementary abelian normal subgroup and $|G/N|$ is prime to its characteristic, the extension splits and its complements are conjugate by $N$. In additive notation an extension cocycle satisfies $f(h,k)+f(hk,t)=hf(k,t)+f(h,kt)$. Averaging over $t$ gives $f(h,k)=b(h)+hb(k)-b(hk)$, where $b(h)=|G/N{|}^{-1}{\sum }_{t}f(h,t)$. Changing the section by $b$ removes the cocycle. For two complements the difference cocycle has $d(hk)=d(h)+hd(k)$; averaging gives $d(h)=b-hb$, which is conjugation by an element of $N$. These arguments use invertibility of the group order on $N$ and underlie Hall conjugacy in soluble groups.

Let $p$ be prime, $a\in {\mathbb{F}}_p^{\times }$, $x,y\in \mathbb{Z}/(p-1)\mathbb{Z}$ and $u,v\in {\mathbb{F}}_p$. Fermat's little theorem makes the displayed multiplication well defined. Its identity is $(0,0)$ and inverse is $(-x,-a^{-x}u)$; associativity follows by expanding both bracketings. Projection onto the first coordinate is a group homomorphism with normal cyclic kernel of order $p$. Under the change $U=a^{-x}u$, multiplication becomes $(x,U)(y,V)=(x+y,U+a^{-x}V)$, the semidirect product of the additive prime field by the cyclic group of order $p-1$. The group is abelian precisely when $a=1$; $a$ need not be a generator of the multiplicative field group.

Let $G=⟨S\mid R⟩$ and $H=⟨T\mid U⟩$ be group presentations, and let $\varphi :H\to \mathrm{Aut}(G)$ be a group homomorphism. Choose a word $w_{t,s}(S)$ representing $\varphi (t)(s)$ for each pair of generators. Then the semidirect product has presentationThe cross-relations allow every word to be written in factor order $gh$, and the multiplication rule matches the prescribed action. The natural homomorphisms to and from the semidirect product are inverse on the generators. Finite factor presentations yield a finite group presentation.

The affine group of the complex line consists of the mapsComposition identifies it with the semidirect product $(\mathbb{C},+)⋊{\mathbb{C}}^{\times }$, where $a$ acts on a translation coordinate by multiplication.

For the Affine group of the complex line,Indeed, the multiplier map $f_{a,b}\mapsto a$ has the displayed translation group as its kernel and abelian image, whileproduces every translation.

The restricted wreath product of groups $H$ and $G$ iswhere $G$ acts on the finitely supported functions $G\to H$ by translation.

The base group acts independently on each coordinate of a Cartesian power, and the top group permutes coordinates. This is distinct from the imprimitive action on a disjoint union of blocks. It is primitive when the base action is primitive and nonregular and the top group is transitive. The full product-action overgroup is $S_{|\Delta |}\wr S_r$. Socles of almost-simple, diagonal or holomorph bases produce respectively product-action, compound-diagonal or holomorph-compound types.

If a group $K$ acts on a finite set $X$, it acts on the direct product of groups $H^X$ by permuting its factors. The resulting semidirect product is the permutation wreath product. For $K=S_m$ acting on $m$ two-point blocks and $H=C_2$, it describes independent within-block swaps combined with a permutation of blocks, and has order $2^{m}m!$. The action on $X$ is part of the definition; it need not be the regular action used in a regular wreath product.

For any perfect group $H$ and $m\ge 3$, the permutation wreath product $H\mathrm{wr}{A}_m$ is generated by elements of order three. The top $A_m$ is generated by 3-cycles. With $\sigma =(123)$, the product of the conjugate $h_1\sigma h_1^{-1}$ and ${\sigma }^{-1}$ is $h_1{h}_2^{-1}$; using ${\sigma }^{-1}$ gives $h_1{h}_3^{-1}$. The commutator of $h_1{h}_2^{-1}$ and $k_1{k}_3^{-1}$ is $[h,k{]}_1$, since distinct coordinate groups commute. Perfectness gives the entire first coordinate group, and transitivity gives all coordinates. No finite-generation assumption on $H$ is needed.

For every $m\ge 5$, let $A=A_m$ act by rooted permutations on the rooted tree of finite words over $m$ letters. Define $b(a)$ to have local permutation $a\in A$ at the pairwise incomparable vertices $1^{r}2$ for $r\ge 0$, and identity elsewhere. Then $b$ is an embedding of $A$ and satisfies $b(a)=(b(a),a,1,\dots ,1)$ with trivial root permutation. The group $H=⟨A,b(A)⟩$ is finitely generated and perfect because it is generated by two finite perfect subgroups.

The root permutation and sections define an injective homomorphism $H\to H\mathrm{wr}{A}_m$. To prove surjectivity, conjugate $b(c)$ by the rooted cycle $(234)$ and take its commutator with $b(a)$. In wreath coordinates this leaves only $b([a,c])$ in the first coordinate. Perfectness of $A_m$ therefore supplies the entire localized copy of $b(A)$ there. Canceling it from the recursion for $b(a)$ supplies a localized rooted $a$ in the second coordinate. Transitivity moves both kinds of generators to every coordinate, yielding the full base $H^m$ and hence the full wreath product. The group is infinite, since a nontrivial finite order $h$ could not satisfy $h=h^m|A_m|$.

The lamplighter group is the restricted wreath product $C_2\wr \mathbb{Z}$. It is generated by one lamp switch and the translation of the integer line.

Let $B\in {\mathrm{GL}}_d(\mathbb{Z})$ and ${\Gamma }_B={\mathbb{Z}}^d{⋊}_B\mathbb{Z}$. If $t$ generates the second factor, then the elements $tn{t}^{-1}{n}^{-1}$ fill $(B-I){\mathbb{Z}}^d$, and henceIf $1$ is not an eigenvalue of $B$, the second summand is finite.

The one-dimensional affine group over ${\mathbb{F}}_q$ consists of maps $x\mapsto ax+b$ with $a\ne 0$.

A nonidentity involution in the affine group over a finite field ${\mathbb{F}}_{\ell }$, for odd prime $\ell$, has linear coefficient $-1$: a translation has order $\ell$, while $u^2=1$ forces $u=\pm 1$. Its unique fixed point is $b/2$, and its remaining orbits have size two. This orbit count computes primes in an intermediate field as double cosets for an order-two decomposition group acting on the radical roots.

// Target: number-theory.bigb

For distinct or equal prime numbers $\ell ,\lambda$, the polynomial $X^{\ell }-\lambda$ is Eisenstein at $\lambda$. Its splitting field over $\mathbb{Q}({\zeta }_{\ell })$ has Galois group a subgroup of $C_{\ell }$, which is nontrivial because a degree-$\ell$ radical cannot belong to a degree-$(\ell -1)$ cyclotomic field. Thus the full Galois group has order $\ell (\ell -1)$. On roots indexed by $j\in {\mathbb{F}}_{\ell }$, all automorphisms act as $j\mapsto uj+b$. Equality of orders identifies the group with the full affine group over a finite field.

// Target: algebra.bigb

For an odd prime $p$, the splitting fieldof $X^p-2$ has Galois groupThe translation subgroup multiplies $2^{1/p}$ by powers of ${\zeta }_p$, while the multiplicative subgroup acts on ${\zeta }_p$ by cyclotomic automorphisms.

Let $Q$ be the subgroup of nonzero squares in ${\mathbb{F}}_{11}^{\times }$. The affine mapsform the nonabelian semidirect product $C_{11}⋊C_5$, with multiplication

This group has seven conjugacy classes: the identity; two classes of five nonidentity translations, distinguished by square class of the translation parameter; and, for each of the four $r\in Q\setminus \{1\}$, one class $\{(r,b):b\in {\mathbb{F}}_{11}\}$ of size eleven.