Question

Let \(H=\left\\{\phi \in S_{n} \mid \phi(n)=n\right\\}\). Find the index of \(H\) in \(S_{n}\).

Short answer

The index of \(H\) in \(S_n\) is \(n\).

Step-by-step solution

Understand the Problem

We are given a subgroup \(H\) of the symmetric group \(S_n\), which consists of all permutations \(\phi\) that fix the element \(n\). Our task is to find the index of this subgroup in \(S_n\). This index is the number of distinct left cosets of \(H\) in \(S_n\).

Calculate the Order of \(S_n\)

The symmetric group \(S_n\) consists of all permutations of \(n\) elements. The number of such permutations, or the order of the group \(S_n\), is \(n!\).

Determine the Order of \(H\)

The subgroup \(H\) consists of all permutations fixing the element \(n\). This is equivalent to permuting the remaining \(n-1\) elements, so \(H\) is isomorphic to \(S_{n-1}\). Thus, the order of \(H\) is \((n-1)!\).

Compute the Index

The index of \(H\) in \(S_n\) is given by the ratio of their orders: \[ \text{Index of } H = \frac{|S_n|}{|H|} = \frac{n!}{(n-1)!} = n. \]

Conclusion

The index of the subgroup \(H\) in the symmetric group \(S_n\) is equal to \(n\), which means there are \(n\) distinct permutations of \(n\) that result in different cosets of \(H\).

Key concepts

Permutations

In mathematics, a permutation of a set is a specific way of arranging the elements of that set. When dealing with permutations, we usually focus on the symmetric group, denoted as \(S_n\), which comprises all possible permutations of \(n\) elements. For example, if \(n = 3\), the set \{1, 2, 3\} can be permuted as (1,2,3), (1,3,2), (2,1,3), (2,3,1), (3,1,2), and (3,2,1), yielding a total of \(3! = 6\) permutations.

Permutations can be written in different notations. One common way is the two-line notation, where the top row represents the original positions and the bottom row the new positions. For instance, a permutation \(\sigma\):
\[\sigma = \begin{pmatrix} 1 & 2 & 3 \ 3 & 1 & 2 \end{pmatrix} \]
means 1 goes to position 3, 2 goes to position 1, and 3 goes to position 2. These permutations form a group because they can be combined and have an inverse such that the composition of a permutation and its inverse is the identity permutation.

This framework is fundamental when understanding how elements can be rearranged, especially within the context of solving problems in the symmetric group \(S_n\), like determining indices and cosets.

Subgroup

A subgroup is a subset of a group that is itself a group under the same operation defined on the original group. For example, in the problem we are discussing, the subgroup \(H\) consists of permutations from \(S_n\) where one specific element \(n\) remains unchanged.

To be classified as a subgroup, a set must fulfill several conditions:

• It must include the identity element of the parent group.
• Each element in the subgroup must have an inverse that is also within the subgroup.
• The composition (or product) of any two elements from the subgroup must still be within the subgroup.

In our problem, \(H\) mirrors these properties since it contains all permutations leaving \(n\) fixed, allowing permutations of the remaining \(n-1\) elements. This means that any arrangement of those \(n-1\) elements is still within \(H\), making \(H\) a valid subgroup, isomorphic to \(S_{n-1}\). Understanding subgroups helps simplify complex group structures by focusing on smaller, manageable parts while preserving the group operations.

Index of a Subgroup

The index of a subgroup in a group is an important concept that measures how many copies, or cosets, of the subgroup fit into the full group. Formally, if \(G\) is a group and \(H\) is a subgroup of \(G\), then the index of \(H\) in \(G\), notated as \([G : H]\), is the number of distinct cosets of \(H\) in \(G\).

The index is computed using the formula:
\[ \text{Index of } H = \frac{|G|}{|H|} \]
where \(|G|\) is the order of the whole group, and \(|H|\) is the order of the subgroup.

In our earlier example, the symmetric group \(S_n\) has \(n!\) permutations and the subgroup \(H\) has \((n-1)!\) permutations. Thus the index of \(H\) in \(S_n\) is:

• \(\text{Index of } H = \frac{n!}{(n-1)!} = n\)

This means \(n\) distinct left cosets of \(H\) can be identified in \(S_n\). Understanding the index helps to explore the internal structure of groups, showing how subgroups fit into larger groups.

Cosets

Cosets are a way to partition a larger group into smaller, equivalent parts using a subgroup. In a group \(G\), when you take a subgroup \(H\), by multiplying each element of \(H\) by a fixed element \(g \in G\), you obtain what is known as a coset.

Cosets can be classified as either left or right cosets:

• A left coset of \(H\) in \(G\) for a given element \(g\) is defined as all elements \(g\cdot h\) for each \(h \in H\).
• Similarly, a right coset of \(H\) in \(G\) is defined as all elements \(h\cdot g\) for each \(h \in H\).

The collection of all such distinct cosets divides the group into classes, where each element of \(G\) belongs to exactly one coset.

In the specific problem considered, the index of subgroup \(H\) in \(S_n\) indicates how many distinct cosets there are, essentially partitioning \(S_n\) into \(n\) parts. Each coset represents a different way that \(H\) aligns within the group, demonstrating different arrangements and combinations of group elements.