Semidirect product

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

Twisted cyclic pair group ( $(x, u) (y, v) = (x + y, a^{y} u + v)$ )

Let

p

be prime,

a \in F_{p}^{\times}

x, y \in Z / (p - 1) Z

and

u, v \in 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.

Presentation of a semidirect product

 0  0

Let

G = ⟨ S ∣ R ⟩

and

H = ⟨ T ∣ U ⟩

be group presentations, and let

ϕ : H \to Aut (G)

be a group homomorphism. Choose a word

w_{t, s} (S)

representing

ϕ (t) (s)

for each pair of generators. Then the semidirect product has presentation

G ⋊_{ϕ} H = ⟨ S ⊔ T ∣ R, U, t s t^{- 1} = w_{t, s} (t \in T, s \in S)⟩ .

(1)

The cross-relations allow every word to be written in factor order

g h

, 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.

Affine group of the complex line ( $Aff (C)$ )

 0  0

The affine group of the complex line consists of the maps

f_{a, b} (z) = a z + b, a \in C^{\times}, b \in C .

(1)

Composition identifies it with the semidirect product

(C, +) ⋊ C^{\times}

, where

a

acts on a translation coordinate by multiplication.

Commutator subgroup of the affine group of the complex line

 0  0

For the Affine group of the complex line,

[G, G] = {f_{1, b} : b \in C} ≅ (C, +), G / [G, G] ≅ C^{\times} .

(1)

Indeed, the multiplier map

f_{a, b} \mapsto a

has the displayed translation group as its kernel and abelian image, while

[f_{a, 1}, f_{1, b}] = f_{1, b (1 - a^{- 1})}

(2)

produces every translation.

Permutation wreath product ( $H ≀_{X} K = H^{X} ⋊ K$ )

 0  0

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.

Lamplighter group ( $C_{2} ≀ Z$ )

 0  0

The lamplighter group is the restricted wreath product

C_{2} ≀ Z

. It is generated by one lamp switch and the translation of the integer line.

Abelianization of a semidirect product by the integers

 0  0

Let

B \in GL_{d} (Z)

and

Γ_{B} = Z^{d} ⋊_{B} Z

. If

t

generates the second factor, then the elements

t n t^{- 1} n^{- 1}

fill

(B - I) Z^{d}

, and hence

Γ_{B}^{ab} ≅ Z \oplus Z^{d} / (B - I) Z^{d} .

(1)

1

is not an eigenvalue of

B

, the second summand is finite.

Affine group over a finite field

 0  0

The one-dimensional affine group over

F_{q}

consists of maps

x \mapsto a x + b

with

a \neq = 0

Affine Galois group of the splitting field of x to the p minus two

 0  0

For an odd prime

p

, the splitting field

Q (ζ_{p}, 2^{1/ p})

(1)

X^{p} - 2

has Galois group

C_{p} ⋊ F_{p}^{\times} ≅ AGL_{1} (F_{p}) .

(2)

The translation subgroup multiplies

2^{1/ p}

by powers of

ζ_{p}

, while the multiplicative subgroup acts on

ζ_{p}

by cyclotomic automorphisms.

Affine semidirect product of cyclic groups of orders eleven and five

 0  0

Let

Q

be the subgroup of nonzero squares in

F_{11}^{\times}

. The affine maps

x ⟼ r x + b, r \in Q, b \in F_{11},

(1)

form the nonabelian semidirect product

C_{11} ⋊ C_{5}

, with multiplication

(r, b) (s, c) = (rs, b + rc) .

(2)

Conjugacy classes in the affine semidirect product of orders eleven and five

 0  0

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 ∖ {1}

, one class

{(r, b) : b \in F_{11}}

of size eleven.

 Discussion (0)

New discussion

There are no discussions about this article yet.

 Articles by others on the same topic (2)

Semidirect product by

Wikipedia Bot  1

 View more

The semidirect product is a construction in group theory that allows you to combine two groups in a specific way, providing a means to build new groups from known ones. It is particularly useful in the study of groups with a certain structure and in applications in various areas of mathematics, including geometry and physics.

 Read the full article

Semidirect product by

Ciro Santilli  40 Updated 2025-07-16

 View more

As per en.wikipedia.org/w/index.php?title=Semidirect_product&oldid=1040813965#Properties, unlike the Direct product, the semidirect product of two goups is neither unique, nor does it always exist, and there is no known algorithmic way way to tell if one exists or not.

This is because reaching the "output" of the semidirect produt of two groups requires extra non-obvious information that might not exist. This is because the semi-direct product is based on the product of group subsets. So you start with two small and completely independent groups, and it is not obvious how to join them up, i.e. how to define the group operation of the product group that is compatible with that of the two smaller input groups. Contrast this with the Direct product, where the composition is simple: just use the group operation of each group on either side.

Product of group subsets

So in other words, it is not a function like the Direct product. The semidiret product is therefore more like a property of three groups.

The semidirect product is more general than the direct product of groups when thinking about the group extension problem, because with the direct product of groups, both subgroups of the larger group are necessarily also normal (trivial projection group homomorphism on either side), while for the semidirect product, only one of them does.

Conversely, en.wikipedia.org/w/index.php?title=Semidirect_product&oldid=1040813965 explains that if

G = N ⋊ H

, and besides the implied requirement that N is normal, H is also normal, then

G = N \times H

Smallest example:

D_{6} = C_{3} ⋊ C_{2}

where

D

is a dihedral group and

C

are cyclic groups.

C_{3}

(the rotation) is a normal subgroup of

D_{6}

, but

C_{2}

(the flip) is not.

Note that with the Direct product instead we get

C_{6}

and not

D_{6}

, i.e.

C_{3} \times C_{2} = C_{6}

as per the direct product of two cyclic groups of coprime order is another cyclic group.

TODO:

why does one of the groups have to be normal in the definition?
what is the smallest example of a non-simple group that is neither a direct nor a semi-direct product of any two other groups?

Bibliography: math.stackexchange.com/questions/1726939/is-this-intuition-for-the-semidirect-product-of-groups-correct

 Read the full article

  See all articles in the same topic Create my own version

Semidirect product

Twisted cyclic pair group ( $(x, u) (y, v) = (x + y, a^{y} u + v)$ )

Presentation of a semidirect product

Affine group of the complex line ( $Aff (C)$ )

Commutator subgroup of the affine group of the complex line

Wreath product ( $H ≀ G$ )

Permutation wreath product ( $H ≀_{X} K = H^{X} ⋊ K$ )

Lamplighter group ( $C_{2} ≀ Z$ )

Abelianization of a semidirect product by the integers

Affine group over a finite field

Affine Galois group of the splitting field of x to the p minus two

Affine semidirect product of cyclic groups of orders eleven and five

Conjugacy classes in the affine semidirect product of orders eleven and five

 Ancestors (5)

 Incoming links (27)

 Discussion (0)

 Articles by others on the same topic (2)

 Discussion (0)  Subscribe (1)

 Discussion (0)