# Subgroups of Symmetric and Dihedral groups

• Avatrin
In summary, the speaker is experiencing difficulty working with subgroups in symmetric and dihedral groups. Specifically, they struggle with showing a subgroup is normal or not, finding a subgroup of a specific order, and proving the non-existence of a subgroup of a certain order. They are seeking strategies and exercises to improve their understanding in this area.
Avatrin
I am having problem working with the objects in the title. Working with permutations, rotations and reflections is fine, but I have problem with the following:

Showing a subgroup is or is not normal (usually worse in the case of symmetric groups)

Finding a subgroup of order n.

Showing that there is no subgroup of order n.

I cannot remember encountering many exercises that helped me learn to work with subgroups of symmetric and dihedral groups. Are there any strategies I can follow, and, even better, any sets of exercises anybody here recommends?

## 1. What is a subgroup of a symmetric group?

A subgroup of a symmetric group is a subset of the group's elements that forms a group itself. In other words, it is a smaller group within a larger group.

## 2. What is the order of a subgroup of a symmetric group?

The order of a subgroup of a symmetric group is the number of elements in the subgroup. It is always a factor of the order of the original symmetric group.

## 3. How are dihedral groups related to symmetric groups?

Dihedral groups are a type of subgroup of symmetric groups. They are formed by the symmetries of regular polygons, and their elements are rotations and reflections of the polygon.

## 4. Can a subgroup of a symmetric group be isomorphic to the original group?

Yes, it is possible for a subgroup of a symmetric group to be isomorphic to the original group. This means that the subgroup has the same structure and properties as the original group, but with different elements.

## 5. How are subgroups of symmetric groups useful in mathematics?

Subgroups of symmetric groups are useful in various areas of mathematics, including group theory, geometry, and combinatorics. They allow for the study and classification of different types of symmetries and patterns within a group.

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