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Chapter 10: Problem 8

If \(G\) is cyclic, prove that \(G / H\) must also be cyclic.

Short Answer

Expert verified
Question: Prove that if a group G is cyclic and H is a subgroup of G, then the quotient group G/H is also cyclic. Answer: To prove that the quotient group G/H is cyclic, we need to show that there exists an element in G/H that can generate the entire quotient group. By selecting the coset xH = gH, where x = g is the generator of G, we show that every coset yH can be expressed as a power of the coset gH. Since every coset yH belongs to the cyclic group generated by gH, we can conclude that G/H = , and therefore, the quotient group G/H is cyclic.

Step by step solution

01

Definition of cyclic group

A group \(G\) is cyclic if there exists an element \(g \in G\) such that every element in the group can be expressed as a power of \(g\). In this case, we say that \(G = \langle g \rangle\). Similarly, if \(G/H\) is cyclic, there exists an element \(xH \in G/H\) such that \(G/H = \langle xH \rangle\).
02

Select an appropriate generator

Let \(g\) be the generator of the cyclic group \(G\). Select the coset \(xH = gH\), where \(x = g\) is the generator of \(G\). To show that \(G/H\) is cyclic, we need to prove that \(G/H = \langle gH \rangle\).
03

Calculate powers of the coset

Consider the powers of \(gH\) in the quotient group \(G/H\): \((gH)^1 = gH\) \((gH)^2 = g^2H\) \((gH)^n = g^nH\) Since \(G\) is a cyclic group, every element in the group can be expressed as a power of \(g\), i.e., for any element \(y \in G\), there exists an integer \(n\) such that \(y = g^n\).
04

Show that \(yH \in \langle gH \rangle\)

We want to show that for every element \(y \in G\), the coset \(yH\) belongs to \(\langle gH \rangle\). Since \(y = g^n\), we can write \(yH\) as a multiple of the coset \(gH\): \(yH = (g^n)H = (gH)^n\) From our earlier calculations, we know that \((gH)^n\) is an element of the cyclic group generated by \(gH\).
05

Conclude that \(G/H\) is cyclic

Since we have shown that every coset \(yH\) can be expressed as a power of the coset \(gH\), we can conclude that \(G/H = \langle gH \rangle\), meaning that the quotient group \(G/H\) is cyclic, as required.

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Key Concepts

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

Group Theory
In mathematics, group theory studies the algebraic structures known as groups, which are collections of elements combined by a binary operation satisfying certain properties. The primary properties include associativity, identity, and invertibility. These concepts form the foundation of group theory.
  • Associativity means that any grouping of operations will result in the same outcome, for instance, \((a imes b) imes c = a imes (b imes c)\).
  • Identity element exists such that for any element \(a\) in the group, \(a imes e = e imes a = a\).
  • Inverses are present for every element \(a\), so there is an element \(b\) such that \(a imes b = b imes a = e\), where \(e\) is the identity element.

Each group possesses a unique structure, whether finite or infinite, abelian (where elements commute) or non-abelian. Understanding these structures assists in analyzing the symmetries and behaviors of mathematical systems, making group theory a crucial component in various mathematical fields like geometry and number theory.
Quotient Groups
Quotient groups are a fascinating concept within group theory, generated by dividing a group \(G\) by one of its normal subgroups \(H\). This process forms the quotient group \(G/H\), a simpler group that retains some of the original properties of \(G\) while providing new insights.
  • The normal subgroup \(H\) allows the group to be partitioned into disjoint sets called cosets.
  • Each element of the quotient group is one of these cosets, symbolized by \(gH\) for an element \(g\) of \(G\).

Quotient groups simplify computations by reducing the complexity of the group \(G\), focusing instead on how the elements outside \(H\) relate to \(H\) itself. In the context of cyclic groups, if \(G\) is cyclic, the quotient group \(G/H\) often maintains a cyclic structure, simplifying further analysis and revealing the underlying symmetries and patterns.
Generators
Generators play an essential role in defining the structure of groups, particularly in cyclic groups, where a single generator can represent every element through its powers. This means that every element in the group \(G\) can be expressed as \(g^n\) for some integer \(n\), where \(g\) is the generator.
  • The generator encapsulates the entirety of the group鈥檚 structure within one element.
  • In the context of quotient groups, powers of this generator, like \(gH\), form the basic building blocks of \(G/H\).

Generators simplify group representation, making it easier to compute within the group. For instance, in proving that the quotient group \(G/H\) is cyclic, identifying \(gH\) as a generator shows that every element in \(G/H\) can be constructed as a power of \(gH\). This greatly enhances our ability to understand and manipulate the group structures, revealing symmetries or simplifying problems associated with algebraic solutions.

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Most popular questions from this chapter

Show that the intersection of two normal subgroups is a normal subgroup.

Find all the subgroups of the quaternion group, \(Q_{8}\). Which subgroups are normal? What are all the factor groups of \(Q_{8}\) up to isomorphism?

For each of the following groups \(G,\) determine whether \(H\) is a normal subgroup of \(G\). If \(H\) is a normal subgroup, write out a Cayley table for the factor group \(G / H\). (a) \(G=S_{4}\) and \(H=A_{4}\) (b) \(G=A_{5}\) and \(H=\\{(1),(123),(132)\\}\) (c) \(G=S_{4}\) and \(H=D_{4}\) (d) \(G=Q_{8}\) and \(H=\\{1,-1, I,-I\\}\) (e) \(G=\mathbb{Z}\) and \(H=5 \mathbb{Z}\)

Let \(H\) be a subgroup of index 2 of a group \(G\). Prove that \(H\) must be a normal subgroup of \(G\). Conclude that \(S_{n}\) is not simple for \(n \geq 3\).

Find all the subgroups of \(D_{4}\). Which subgroups are normal? What are all the factor groups of \(D_{4}\) up to isomorphism?
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