# Show a group is a semi direct product

• DeldotB
In summary, a semi direct product is a mathematical structure that combines two groups while preserving their individual structures. To show that a group G is a semi direct product, a homomorphism must exist between two subgroups H and K. The homomorphism must be injective and satisfy the semi direct product property. A group can have multiple semi direct product structures, and they have applications in various fields of science.
DeldotB

## Homework Statement

Good day,

I need to show that $S_n=\mathbb{Z}_2$(semi direct product)$Alt(n)$
Where $S_n$ is the symmetric group and $Alt(n)$ is the alternating group (group of even permutations) note: I do not know the latex code for semi direct product

none

## The Attempt at a Solution

(I think) It suffices to show that $\mathbb{Z}_2\cap Alt(n)=0$ (where 0 is the identity) and that
$S_n= \mathbb{Z}_2 Alt(n)$. My question is, is $S_n= \mathbb{Z}_2 Alt(n)$ even valid notation? And how do I begin to do this? p.s I get this notation from my book which says:

To show a group G is a semi direct product, show $G=NH$ and $N \cap H$= identity.

I should mention here that the alternating group is the normal subgroup (I think).

For the first part, since $\mathbb{Z}_2$= {0,1}, its pretty clear that its intersection with Alt(n) is just the identity. I am having problems with the second part..

The latex code for semidirect product is \rtimes.

The problem's notation is muddled. It should ask you to prove either

(1) isomorphism, not equality, ie to prove that ##S_n\cong \mathbb{Z}_2\rtimes Alt(n)##, where the semidirect product is Outer.
OR
(2) equality, where the semidirect product is Inner, ie ##S_n=N\rtimes Alt(n)## where ##N## is any subgroup of ##S_n## of order 2, which hence must be isomorphic to ##\mathbb{Z}_2##.

I suggest trying for the second one.

DeldotB said:
It suffices to show that ##\mathbb{Z}_2\cap Alt(n)=0## (where 0 is the identity) and that
##S_n= \mathbb{Z}_2 Alt(n)##. My question is, is ##S_n= \mathbb{Z}_2 Alt(n)## even valid notation?
The notation is valid. If ##G,H## are subgroups ##GH## is defined as the set of all elements that can be written as ##gh## where ##g\in G,\ h\in H##. That is not necessarily a subgroup. So part of what you have to show is that ##S_n= \mathbb{Z}_2 Alt(n)## (or rather ##N\,Alt(n)## using my notation of (2) above) is a subgroup.

Why not pick N to be the subgroup generated by the swap permutation (1 2). Then try to prove the two things you need to prove. The intersection one is dead easy. The other, not so much.

## 1. What is a semi direct product?

A semi direct product is a mathematical structure that combines two groups in a way that preserves the individual structures of each group. It is a generalization of the direct product, where the two groups are combined in a more flexible manner.

## 2. How do you show that a group is a semi direct product?

To show that a group G is a semi direct product of two subgroups H and K, you need to demonstrate that G can be written as a product of H and K, and that there exists a homomorphism from K to the automorphism group of H that satisfies certain conditions.

## 3. What are the conditions for a homomorphism to exist between two groups in a semi direct product?

The homomorphism from K to the automorphism group of H must be injective and must satisfy the condition that the image of each element of K commutes with the corresponding element of H. This is known as the semi direct product property.

## 4. Can a group have multiple semi direct product structures?

Yes, it is possible for a group to have multiple semi direct product structures. This is because there can be different ways to combine two subgroups that satisfy the semi direct product property. However, there may be cases where only one semi direct product structure exists for a given group.

## 5. What are the applications of semi direct products in science?

Semi direct products have applications in many areas of science, including physics, chemistry, and computer science. They are particularly useful in studying symmetries and transformations in physical systems, and in designing efficient algorithms for solving computational problems.

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