# Isomorphism of dihedral with a semi-direct product

• AllRelative
In summary, the conversation discusses a problem involving groups and the goal is to show that Dm is isomorphic to the semidirect product of Zm and Z2, with a specific function phi. The conversation includes attempts at solving the problem and the suggestion to write it in terms of elements r and s. It is also mentioned that the function phi needs to be defined for both Zm and Z2.
AllRelative

## Homework Statement

Let m ≥ 3. Show that $$D_m \cong \mathbb{Z}_m \rtimes_{\varphi} \mathbb{Z}_2$$
where $$\varphi_{(1+2\mathbb{Z})}(1+m\mathbb{Z}) = (m-1+m\mathbb{Z})$$

## Homework Equations

I have seen most basic concepts of groups except group actions. Si ideally I should not use them for this problem.

## The Attempt at a Solution

So I've been thinking about this problem for a couple of days and I just can't seem to arrive to the proof I am looking for.

And so we have that <s> is cyclical of order 2 and <r> is cyclical of order m. Therefore, $$\langle s \rangle \cong \mathbb{Z}_2$$
and $$\langle r \rangle \cong \mathbb{Z}_m$$
I feel I have to use the fact that <r> is normal in Dm and that <r>∩<s> = {e}. I unsure where to go from there

Thanks for the help

I suggest to write it in terms of ##r## and ##s## first, and then see how it translates to ##\mathbb{Z}_m##, resp. ##\mathbb{Z}_2##. And I will write the representatives, aka elements of ##\mathbb{Z}_m## as ##[k] = k+m\mathbb{Z}## which is more convenient and easier to read.

In general a semidirect product ##G = N \rtimes_\varphi H## goes by ##(n_1\, , \,h_1)\cdot (n_2\, , \,h_2) \stackrel{(*)}{=} (n_1 \cdot \varphi(h_1)(n_2)\, , \,h_1\cdot h_2)##, see e.g. https://en.wikipedia.org/wiki/Semidirect_product. We have ##\varphi\, : \,H \longrightarrow \operatorname{Aut}(N)##.

Here we have products ## P =(r^k\; , \;s^\varepsilon)\cdot (r^n\; , \;s^\eta) = r^k\cdot s^\varepsilon \cdot r^n \cdot s^\eta## with ##\varepsilon \in \{\,0,1\,\}##.
Now write ##P## in the form ##P=r^l \cdot s^\mu=(r^l,s^\mu)## and compare that with ##(*)## to see how ##\varphi(s^\varepsilon)(r^n)## has to be defined.
If you're done, you can write this as ##\varphi([\varepsilon])([n])##.

You have only defined ##\varphi([1])([1])##. What are ##\varphi([0])## and ##\varphi([1])([n])## if ##n>1## or short: What is ##\varphi([\varepsilon])## with ##\varepsilon \in \{\,0,1\,\} = \{\,[0],[1]\,\}\,?##

## 1. What is an isomorphism?

Isomorphism is a mathematical concept that describes a relationship between two mathematical structures that preserves their essential properties. In simpler terms, it means that two structures are essentially the same even though they may look different.

## 2. What is dihedral?

Dihedral refers to a group of symmetries that can be applied to a regular polygon, such as a square or equilateral triangle. It is represented by the symbol Dn, where n is the number of sides of the polygon.

## 3. What is a semi-direct product?

A semi-direct product is a mathematical operation that combines two groups in a specific way to form a new group. It is denoted by the symbol ⋉ and is used to describe the structure of certain groups, such as the dihedral group Dn.

## 4. How is dihedral related to a semi-direct product?

The dihedral group Dn can be expressed as a semi-direct product of two groups - a cyclic group of rotations and a cyclic group of reflections. This means that the structure and properties of Dn can be understood through the structure and properties of these two groups.

## 5. Why is the isomorphism of dihedral with a semi-direct product important?

The isomorphism between dihedral and a semi-direct product allows us to better understand and analyze the properties of the dihedral group. It also helps us to connect dihedral with other mathematical concepts and structures, making it a useful tool in various fields of mathematics and science.

• Calculus and Beyond Homework Help
Replies
7
Views
1K
• Calculus and Beyond Homework Help
Replies
1
Views
1K
• Calculus and Beyond Homework Help
Replies
5
Views
1K
• Calculus and Beyond Homework Help
Replies
5
Views
1K
• Calculus and Beyond Homework Help
Replies
3
Views
1K
• Calculus and Beyond Homework Help
Replies
9
Views
2K
• Calculus and Beyond Homework Help
Replies
13
Views
3K
• Calculus and Beyond Homework Help
Replies
5
Views
861
• Calculus and Beyond Homework Help
Replies
3
Views
1K
• Calculus and Beyond Homework Help
Replies
1
Views
1K