In Z, let H=⟨5⟩and K=⟨7⟩. Prove that Z=H K. Does Z=H ×K ? |...

Name: Problem 34, Chapter 9: Contemporary Abstract Algebra
Uploaded: 2021-07-27T13:33:55-08:00
Duration: 7 min 57 s
Channel: Sandip Ranjan
Description: In Z, let H=⟨5⟩and K=⟨7⟩. Prove that Z=H K. Does Z=H ×K ?

Key Concepts

Direct Product of Groups

The direct product of two groups is a construction where pairs of elements from each group are combined. This product forms a new group in which the group operation is performed componentwise. For an external direct product, each element is uniquely represented as a pair, making the structure very different from a cyclic group like Z unless the subgroups involved intersect trivially.

Internal Direct Product

An internal direct product occurs when a group can be decomposed into a product of two subgroups whose elements combine to give every element of the original group and whose intersection is only the identity. In the context of Z with subgroups generated by 5 and 7, although the sum of these subgroups yields all of Z, their intersection is nontrivial, meaning Z is not the internal direct product of these two subgroups.

Subgroup Product

The product of two subgroups (or, in abelian groups like Z, their sum) is the set of all elements formed by adding an element from one subgroup to an element from the other. Showing that every element of Z can be written as a sum from the subgroups generated by 5 and 7 involves using linear combinations, a consequence of Bézout's identity when the generators are relatively prime.

Cyclic Groups

A cyclic group is one that can be generated by a single element, meaning every element of the group can be written as some multiple (or power) of that generator. The group of integers Z under addition is a standard example of an infinite cyclic group, and its subgroups, such as those generated by 5 or 7, are also cyclic.

Subgroup Generation

Subgroup generation is the process of creating a subgroup from a given element by taking all its integral multiples (or powers). In Z, the subgroup generated by an integer 'n' consists of all multiples of 'n'. This concept is essential in understanding the structure of groups like Z and its various subgroups.

Bézout's Identity

Bézout's Identity states that if a and b are integers with greatest common divisor d, then there exist integers x and y such that ax + by = d. When a and b are relatively prime (d = 1), this identity shows that every integer can be expressed as a linear combination of a and b. This key idea explains why the sum of the subgroups generated by 5 and 7 covers the entire group Z.

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