Determining Quotient Group (\mathbb{Z}_2\times\mathbb{Z}_4)/\langle(1,2)\rangle

Mar 29, 2005

Oxymoron

Ok I may have something.

Dihedral groups are special groups that consist of rotations [tex]a[/tex] and relfections [tex]b[/tex], where the group operation is the composition of these rotations and reflections.

The finite dihedral group [tex]\mathcal{D}_n[/tex] has [tex]2n[/tex] elements and is generated by [tex]a[/tex] (with order [tex]n[/tex]) and [tex]b[/tex] (with order [tex]2[/tex]). The two elements of the dihedral group satsify

[tex]ab = ba^{-1}[/tex]

If the order of [tex]\mathcal{D}_{n}[/tex] is greater than 4, then the group operations do not commute, ie [tex]\mathcal{D}_n[/tex] is not abelian.

The [tex]2n[/tex] elements of [tex]\mathcal{D}_n[/tex] are

[tex]\{e, a, a^2, \dots , a^{n-1}, b, ba, ba^2, \dots , ba^{n-1}\}[/tex]

Now, in order to form the quotient groups, I need to find the normal subgroups. The normal subgroups are those which are invariant under conjugation.

I know [tex]\{e\}[/tex] and [tex]\{\mathcal{D}_n\][/tex] are going to normal subgroups. But I don't know how to find any others.

2. How do you determine the quotient group (\mathbb{Z}_2\times\mathbb{Z}_4)/\langle(1,2)\rangle?

To determine the quotient group, we first need to find all the cosets of the subgroup \langle(1,2)\rangle within the original group (\mathbb{Z}_2\times\mathbb{Z}_4). Then, we can define the group operation on the cosets and check if it satisfies the group axioms. If it does, then the set of cosets with the defined operation forms the quotient group.

3. What is the significance of the subgroup generated by (1,2) in the quotient group?

The subgroup generated by (1,2) is the kernel of the quotient group. This means that all elements in this subgroup will map to the identity element in the quotient group. It also represents the elements that are "ignored" or "collapsed" in the quotient group, as they do not affect the overall structure of the group.

4. How does the quotient group (\mathbb{Z}_2\times\mathbb{Z}_4)/\langle(1,2)\rangle differ from the original group (\mathbb{Z}_2\times\mathbb{Z}_4)?

The quotient group is a simplified version of the original group, as it only considers the cosets of the subgroup generated by (1,2). This means that the quotient group will have fewer elements and a different group operation compared to the original group. However, both groups will share some similar properties, as the quotient group is derived from the original group.

5. How can the quotient group be used in practical applications?

The quotient group can be used in various applications, such as cryptography, coding theory, and computer science. It allows us to simplify and analyze complex structures, making it easier to solve problems and make predictions. In particular, the quotient group can be used to study the symmetries of objects, which has applications in chemistry, physics, and engineering.