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Chapter 16: Problem 1

If \(H\) is a subgroup of a group \(G\), let \(X\) designate the set of all the cosets of \(H\) in \(G\). For each element \(a \in G\), define \(\rho_{a}: X \rightarrow X\) as follows: $$ \rho_{a}(H x)=H(x a) $$ Prove that each \(\rho_{a}\) is a permutation of \(X\).

Short Answer

Expert verified
Each \(\rho_a\) is a permutation of \(X\) because it is a bijection.

Step by step solution

01

Understand the Problem

We need to show that the function \(\rho_a: X \rightarrow X\) defined by \(\rho_a(Hx) = H(xa)\) is a permutation of the set \(X\) of cosets of \(H\) in \(G\). A permutation is a bijection from a set onto itself, meaning \(\rho_a\) must be both one-to-one and onto.
02

Show That \(\rho_a\) is Onto

To show that \(\rho_a\) is onto, for every \(Hy \in X\), we need to find an element \(Hx \in X\) such that \(\rho_a(Hx) = Hy\). Set \(x = ya^{-1}\), then \(\rho_a(H(ya^{-1})) = H(ya^{-1}a) = Hy\), verifying \(\rho_a\) is onto.
03

Show That \(\rho_a\) is One-to-One

To show that \(\rho_a\) is one-to-one, assume \(\rho_a(Hx_1) = \rho_a(Hx_2)\), which rewrites to \(H(x_1a) = H(x_2a)\). Thus, \(x_1aH = x_2aH\) implies \(x_1H = x_2H\). Thus, \(x_1 \equiv x_2 \pmod{H}\), showing that \(\rho_a\) is indeed one-to-one.
04

Conclusion

Since \(\rho_a\) is both onto and one-to-one, it is a bijection, hence a permutation of the set \(X\). Each element \(a \in G\) therefore defines a permutation \(\rho_a\) on the set of cosets of \(H\) in \(G\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Permutations
A permutation is an important concept in mathematics, especially in group theory. It refers to a rearrangement of the elements of a set in a particular order. In the context of the given problem, a permutation of the set of cosets of subgroup data:def: Different ways are there to rearrange a set's elements.
  • Each time you rearrange it, you're making a permutation.
  • A permutation must rearrange elements such that each usually assumed to be onto and one-to-one.
  • It relates to the idea of transformations of a set where the original and the transformed sets are the same.
  • The set remains the same, but its elements are in different locations.
Here, each function \(\rho_a\) acts on the set of all cosets of \(H\) in \(G\) as a permutation. The function rearranges those cosets without losing any or having duplications. Thus, the set maintains its number of total cosets, but their order can be changed.
Cosets
Cosets are subsets of a group that are created by multiplying a subgroup by a particular element of the group. Consider subgroup \(H\) within group \(G\). A left coset of \(H\) by some element \(g\) is expressed as \(gH\), while a right coset is \(Hg\).
  • Cosets either entirely overlap or are disjoint, depending on the elements used to form them.
  • They help in understanding the larger structure of groups by examining the repeated actions of a subgroup.
  • The collection of all unique cosets form a partition of the group.
In the problem, the set \(X\) consists of all distinct left cosets of \(H\) in \(G\), and the function \(\rho_a\) acts by rearranging these cosets through multiplication by elements in \(G\). Therefore, the concept of cosets is pivotal to defining and working with permutations on this set.
Subgroups
A subgroup is a smaller group contained within a larger group, complying with the group's operation and containing the group's identity element. For a set \(H\) to be a subgroup, it must satisfy the group properties  closure, associativity, the presence of an identity element, and the presence of inverses for each element.
  • Closure means if you perform the group operation on any two elements of \(H\), the result is also in \(H\).
  • The identity of the group must also be in \(H\).
  • Every element must have an inverse within \(H\).
In the given problem, \(H\) is a subgroup of \(G\). By analyzing the set of all cosets of \(H\) in \(G\), we gain insight into the structure and permutations possible within the larger group \(G\). Subgroups like \(H\) assist in simplifying complexity by breaking down groups into manageable sections.
Bijections
A bijection is a special type of function that creates a perfect "one-to-one correspondence" between two sets. Bijections are both injective (one-to-one) and surjective (onto). This means:
  • Each element of the first set is paired with a unique element of the second set, and vice versa.
  • Every element in the second set has a pre-image in the first set.
In the given problem, we needed to show that \(\rho_a\) is a permutation, which entails demonstrating it is a bijection on the set of cosets.
This requires proving it is:- Onto: Every element of the coset set \(X\) has an image in \(X\) through elements of \(G\).- One-to-One: Different elements do not map onto the same coset, ensuring distinctiveness.This bijective property ensures that we have a perfect mapping of the coset set onto itself, confirming \(\rho_a\) as a permutation.

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Most popular questions from this chapter

Let \(G\) be a group; let \(H\) and \(K\) be subgroups of \(G\), with \(H\) a normal subgroup of \(G\). Prove the following : By the FHT, \(H /(H \cap K) \cong H K / H\). (This is referred to as the first isomorphism theorem.)

As a provisional definition, let us call a finite abelian group "decomposable" if there are elements \(a_{1}, \ldots, a_{n} \in G\) such that: (D1) For every \(x \in G\), there are integers \(k_{1}, \ldots, k_{n}\) such that \(x=a_{1}^{k_{1}} a_{2}^{k_{2}} \cdots a_{n}^{k_{n}}\). (D2) If there are integers \(l_{1}, \ldots, l_{n}\) such that \(a_{1}^{l_{1}} a_{2}^{l_{2}} \cdots a_{n}^{l_{n}}=e\) then \(a_{1}^{l_{1}}=a_{2}^{t_{2}}=\cdots\) \(=a_{n}^{l_{n}}=e\) If (D1) and (D2) hold, we will write \(G=\left[a_{1}, a_{2}, \ldots, a_{n}\right]\). Prove: \(G \cong\left\langle a_{1}\right\rangle \times G^{\prime}\). Conclude that \(G \cong\left\langle a_{1}\right\rangle \times\left\langle a_{2}\right\rangle \times \cdots \times\left\langle a_{n}\right\rangle .\)

Let \(G, M\), and \(N\) be groups, let \(f: G \rightarrow M\) be a homomorphism from \(G\) onto \(M\), and let \(h: G \rightarrow N\) be a homomorphism from \(G\) onto \(N\). Show that \(\phi(x)=(f(x), h(x))\) is a homomorphism from \(G\) onto \(M \times N\).

In each of the following, use the fundamental homomorphism theorem to prove that the two given groups are isomorphic. Then display their tables. $$ \mathbb{Z}_{5} \text { and } \mathbb{Z}_{20} /\langle 5\rangle $$

Let \(G\) be a group; let \(H\) and \(K\) be subgroups of \(G\), with \(H\) a normal subgroup of \(G\). Prove the following : Every member of the quotient group \(H K / H\) may be written in the form \(H k\) for some \(k \in K\).
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