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Magazine Chapter 12: Problem 2
Let \(H=\\{I,(12)(34),(13)(24),(14)(23)\\}\). Show that \(H\) is a normal subgroup of \(S_{4}\), so that \(S_{4} / H\) has order six. Write down the six cosets with respect to \(H .\) Is \(S_{4} / H\) is isomorphic to \(C_{6}\) or to \(D_{6}\) ?
Short Answer
Expert verified
Yes, \(H\) is a normal subgroup, and \(S_4 / H\) is isomorphic to \(D_6\).
Step by step solution
01
Verify that H is a subgroup
To show that a set is a subgroup, it must satisfy the identity, closure, and inverse properties. - Identity: The identity element of the symmetric group on 4 elements, denoted by \(I\), is part of \(H\).- Closure: Check each pair of elements in \(H\) to see if their product is also in \(H\). All combinations such as \((12)(34) \cdot (13)(24) = (14)(23)\) are within \(H\).- Inverse: Each element in \(H\) is its own inverse or another element in \(H\), for example, \((12)(34)\cdot (12)(34) = I \). Thus, \(H\) satisfies all group properties.
02
Check normality condition
To check normality, each element \(g\) of \(S_4\) should conjugate elements of \(H\) back into \(H\). For any \(h \in H\) and for any \(g \in S_4\), compute \(ghg^{-1}\). For example, \( (123) (12)(34) (132) = (13)(24) \), which is in \(H\). This holds for all elements in \(S_4\). Hence, \(H\) is a normal subgroup.
03
Determine the order of S4/H
The order of the quotient group \(S_4 / H\) is the order of \(S_4\) divided by the order of \(H\). The order of \(S_4\) (symmetric group of 4 elements) is 24 and the order of \(H\) is 4. Therefore, \(|S_4 / H| = 24 / 4 = 6\).
04
List the cosets of H in S4
The cosets of \(H\) in \(S_4\) are of the form \(gH\). Select representatives \(g\) from \(S_4\). Choosing \(I, (12), (13), (14), (123), (132)\), we find the cosets:- \(H = \{I, (12)(34), (13)(24), (14)(23)\}\)- \((12)H = \{(12), (34), (1324), (1423)\}\)- \((13)H = \{(13), (24), (1234), (1432)\}\)- \((14)H = \{(14), (23), (1243), (1342)\}\)- \((123)H = \{(123), (14)(23), (34), (12)\}\)- \((132)H = \{(132), (13)(24), (12)(34), I\}\).
05
Determine if \(S_{4} / H\) is isomorphic to \(C_6\) or \(D_6\)
The group \(S_4 / H\) has order 6. Compute the properties of this group by observing the structure of its cosets. From the presence of elements involution, \((12)\), and a 3-cyclic permutation, \((123)\), it suggests the structure of a dihedral group matched with elements of order 3 and 2. As such, this group is more complicated than a cyclic group and is isomorphic to \(D_6\), the dihedral group of order 6.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Symmetric Group
The symmetric group, often denoted as \( S_n \), is a fundamental concept in group theory. It consists of all possible permutations of a set of \( n \) elements. For instance, \( S_4 \) comprises all permutations of four elements.
- Identity: Every symmetric group has an identity element, commonly represented as \( I \), which enables no change to the arrangement of elements.
- Order: The order of the symmetric group \( S_n \) is \( n! \) (\( n \) factorial), which accounts for all possible ways to arrange the elements.
- Subgroups: Symmetric groups can have various subgroups, and some of these can be normal subgroups, like \( H \) in the original exercise.
Quotient Group
A quotient group, also called a factor group, is formed by partitioning a group into a set of cosets of a normal subgroup. In simpler terms, every element of the original group is divided into different classes or cosets.
- Normal Subgroup: To form a quotient group, the subgroup must be normal. This means that the group remains invariant under conjugation by elements of the original group.
- Order: The order of the quotient group \( S_4 / H \) is computed by dividing the order of the group by the order of the subgroup. In the exercise, this is calculated as \( 24 / 4 = 6 \).
- Function: Quotient groups help in studying the structure and properties of the original group by examining its simplified version.
Dihedral Group
The dihedral group, denoted as \( D_n \), represents the symmetries of a regular \( n \)-sided polygon, including both rotations and reflections.
- Order: A dihedral group has an order of \( 2n \), accounting for \( n \) rotations and \( n \) reflections.
- Elements: In the case of \( D_6 \), it reflects the symmetries of a hexagon, which align with elements of order 2 (reflections) and elements of order 3 (rotations).
- Isomorphism: In the exercise, \( S_4 / H \) is found to be isomorphic to \( D_6 \), highlighting that it shares the same structural properties and symmetry operations.
Cosets
Cosets are a way of grouping elements of a group into disjoint subsets using a subgroup. Each coset consists of elements formed by multiplying a fixed element from the group with each element of the subgroup.
- Formation: To form a coset, select an element from the original group and multiply it with each element of the subgroup.
- Types: Two types of cosets can be formed - left cosets and right cosets, depending on the order of multiplication with the subgroup.
- Role in Quotient Groups: Cosets are crucial when determining quotient groups since they essentially partition the group into distinct equivalence classes.
