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Chapter 6: Problem 55

Show that \(Z_{p}\) has no proper nontrivial subgroups if \(p\) is a prime number.

Short Answer

Expert verified
Since \(Z_p\) has prime order \(p\), it only has trivial subgroups.

Step by step solution

01

Understand the Definition of Z_p

The group \(Z_p\) is the set of integers \(\{0, 1, 2, \ldots, p-1\}\) under addition modulo \(p\). This means we consider addition within this set such that we 'wrap around' after reaching \(p\).
02

Recognize the Order of Z_p

Since \(Z_p\) consists of the integers from 0 to \(p-1\), it has \(p\) elements. Therefore, the order of \(Z_p\) is \(p\). In group theory, the 'order' of a group is the number of elements in the group.
03

Use Prime Order Property

A known result in group theory is that if a group has prime order \(p\), then it does not have any proper nontrivial subgroups. This means that the only subgroups that can exist are the trivial group \(\{0\}\) and the group itself \(Z_p\).
04

Conclusion

Since \(Z_p\) is a group of prime order, it must, by the prime order property, only have the trivial and itself as subgroups. Therefore, \(Z_p\) has no proper nontrivial subgroups.

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Key Concepts

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

Group Theory
Group theory is a central part of abstract algebra. It studies the algebraic structures known as groups. A group is essentially a set equipped with an operation that combines any two elements to form a third element, satisfying four fundamental properties: closure, associativity, identity, and invertibility.
  • Closure: Performing the group operation on any two group elements gives another element that is still in the group.
  • Associativity: The operation is associative, meaning that the order in which operations are performed does not change the outcome.
  • Identity Element: There is an identity element that, when used in the group operation with any element of the group, leaves the other element unchanged.
  • Invertibility: Every element in the group has an inverse, such that the group operation between this element and its inverse gives the identity element.
Groups can be finite or infinite, and studying them helps understand symmetry, transformations, and even cryptography. In the context of our problem, we are dealing with finite groups.
Subgroups
Subgroups are a crucial concept in group theory. A subgroup is basically a subset of a group that is itself a group under the same operation. To be a subgroup, the subset must follow all the group axioms and it must contain the identity element of the original group.
When exploring subgroups:
  • It's important to check if the operation within the subset is closed.
  • The elements must follow associativity.
  • The identity of the subset must be the same as the identity of the entire group.
  • All elements in the subset must have inverses also within it.
In our exercise, since we are considering the group \(Z_p\), which has a prime order, it is stated that it cannot possess any proper nontrivial subgroups. This means, besides the trivial subgroup \(\{0\}\), which only includes the identity, the whole group is the only other subgroup.
Prime Order
The concept of a group having a prime order is critical in understanding the exercise. A prime number, by definition, has only two distinct positive divisors: 1 and itself. Therefore, when a group is of prime order, it means the total count of its elements is a prime number.
Prime order groups have unique properties:
  • No proper nontrivial subgroups: The only subgroups that can exist are the trivial subgroup containing only the identity and the group itself.
  • All elements, apart from the identity, generate the group: Any non-identity element can be used to produce every other element of the group through repeated application of the group operation.
This property is what resolves our exercise, as it argues that \(Z_p\) cannot have any proper subset that also forms a subgroup except for the trivial and the entire group.
Modulo Operation
The modulo operation is a fundamental concept in number theory and group theory, often represented by '%'. In essence, when performing an operation like addition on elements of \(Z_p\), this operation ensures that the results 'wrap around' once reaching the modulus, \(p\).
The modular arithmetic involved:
  • Enables keeping all computations within a defined set range \([0, p-1]\) for \(Z_p\).
  • Ensures periodicity; numbers restart from zero after reaching the modulus.
  • Preserves group properties like closure under addition.
In \(Z_p\), the set \(\{0, 1, 2, \ldots, p-1\}\), modulo arithmetic guarantees that whatever operation you perform, you stay contained within this set. This makes the modulo operation a pivotal notion in defining the structure and properties of cyclic groups like \(Z_p\).

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Most popular questions from this chapter

The gencrators of the cyclic multiplicative group \(U_{n}\) of all \(n\)th roots of unity in \(\mathbb{C}\) are the primitive \(n\)th roots of unity. In Exercises 38 through 41 , find the primitive \(n\)th roots of unity for the given value of \(n\). $$ n=12 $$

Find all orders of subgroups of the given group. $$ \mathrm{z}_{12} $$

Find the primitive \(n\) roots of unity for the given value of \(n\). $$ n=6 $$

Find the number of elements in the indicated cyctic group. The cyclic subgroup of \(Z_{90}\) generated by 25

In Exercises 1 through 4 , find the quotient and remainder, according to the division algorithm, when \(n\) is divided by \(m\). \(n=-42, m=9\)
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