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$Hol (C_{8})$ is the group of affine maps $x \mapsto a x + b$ with $a \in {1, 3, 5, 7}$

Example

On the additive group $C_{8} = Z / 8$ ,

$Hol (C_{8}) = {x \mapsto a x + b : a \in {1, 3, 5, 7}, b \in Z / 8} .$

These are $32$ distinct permutations.

Facts & Assumptions

Given: The additive cyclic group $C_{8} = Z / 8$ .

[L1]

The holomorph is $G ⋊ Aut (G)$ and acts faithfully by $x \mapsto g α (x)$ ( The holomorph $Hol (G) = G ⋊ Aut (G)$ , The holomorph acts faithfully on $G$ by affine permutations $x \mapsto g α (x)$ ).

[L2]

$Aut (C_{8}) ≅ (Z / 8)^{\times}$ , with a unit $a$ acting by multiplication by $a$ ( $Aut (C_{n}) ≅ (Z / n Z)^{\times}$ ).

[L3]

The units modulo $8$ are exactly the classes represented by integers coprime to $8$ (For $n \geq 1$ , $[a]_{n}$ is a unit if and only if $\gcd (a, n) = 1$ ).

Verification

technique · direct

1.1

By [L3], the four units modulo $8$ are $1, 3, 5, 7$ . Thus [L1] and [L2] identify every holomorph element with one of the displayed affine maps.

L1L2L3

1.2

If $a x + b = c x + d$ for every $x$ , evaluation at $0$ gives $b = d$ , and evaluation at $1$ then gives $a = c$ . Hence the $8 \cdot 4 = 32$ maps are distinct.

algebra

2.1

Their composition is $(x \mapsto a x + b) \circ (x \mapsto c x + d) = x \mapsto a c x + (a d + b)$ , which agrees with the holomorph multiplication from [L1].

L1algebra∎

Depends on

Used by

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Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 68 results over 14 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

Peter J. Cameron, The Holomorph of a Group (standard reference, not scraped)

Hol⁡(C8)Hol(C8​) is the group of affine maps x↦ax+bx↦ax+b with a∈{1,3,5,7}a∈{1,3,5,7}