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Chapter 5: Problem 2

Show that if the left coset \(g H\) is a subgroup of \(G\), then \(g\) is in \(H\).

Short Answer

Expert verified
If \(gH\) is a subgroup, \(g\) must be in \(H\).

Step by step solution

01

Identify Left Coset

A left coset of a subgroup \(H\) in a group \(G\) is the set \(gH = \{gh : h \in H\}\) for a given element \(g \in G\).
02

Subgroup Condition

The problem states that the left coset \(gH\) is a subgroup of \(G\). According to subgroup properties, a subset must include the identity element \(e\) of \(G\) to be a subgroup.
03

Identity in Left Coset

Since \(gH\) is a subgroup, it must contain the identity element. This implies there is an element \(h' \in H\) such that \(gh' = e\).
04

Solve for \(g\)

From \(gh' = e\), solve for \(g\) to find \(g = e(h')^{-1}\). Since \(h'\) is in \(H\) and subgroups are closed under inverse, \((h')^{-1} \in H\), implying \(g \in H\).
05

Conclusion

Since \(g \in H\), we have shown that if \(gH\) is a subgroup of \(G\), then \(g\) must be an element of \(H\).

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Key Concepts

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

Subgroups
In group theory, a _subgroup_ is a special subset of a group that itself forms a group under the same operation as the larger group. For a set to be considered a subgroup, it must satisfy certain properties:
  • Closure: For every pair of elements in the subgroup, their product (under the group operation) must also be in the subgroup.
  • Identity: The subgroup must include the group's identity element.
  • Inverses: Every element in the subgroup must have an inverse that is also in the subgroup.
These properties ensure that a subgroup maintains the structure and rules of the larger group. If the set meets these conditions, we can confidently identify it as a subgroup.
Cosets
Cosets are essentially ways to 'translate' or shift a subgroup within a group. If we have a subgroup \( H \) and a group element \( g \) in a group \( G \), we can form left cosets and right cosets. Let's focus on left cosets here:
  • A left coset \( gH \) of \( H \) is the set \( \{ gh : h \in H \} \), where you multiply each element of \( H \) by \( g \) from the left.
  • Cosets can be thought of as "distinct copies" of the subgroup \( H \), spread throughout the group \( G \).
One interesting property of cosets is that they partition the group into disjoint, equally-sized subsets.
Identity Element
The _identity element_ is a fundamental concept in group theory. It is an element in a group that, when used in a group operation with any other element of the group, leaves the other element unchanged. For a group \( G \) with operation \( \cdot \), the identity element \( e \) has the property:
  • For any element \( a \) in \( G \), we have \( e \cdot a = a \cdot e = a \).
In the context of subgroups, the identity element must be present for a subset to qualify as a subgroup. This presence ensures stability and consistency in the operation and structure of both the subgroup and the entire group.
Group Properties
Groups have specific properties that define their structure and operation, making them an essential component of mathematics and symmetry theory. Here are the key properties:
  • Closure: If you apply the group operation to any elements in the group, the result stays within the group.
  • Associativity: The group operation is associative; combining three elements, \((a \cdot b) \cdot c = a \cdot (b \cdot c) \).
  • Identity Element: There exists an identity element \( e \) in the group such that for any element \( a \), \( a \cdot e = e \cdot a = a \).
  • Inverses: Every element \( a \) in the group has an inverse \( a^{-1} \) such that \( a \cdot a^{-1} = a^{-1} \cdot a = e \).
These properties not only define group structure but also ensure that group-related mathematical concepts, like subgroups and cosets, can function within this framework with consistency.

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Most popular questions from this chapter

Give an example of a group \(G\) with subgroups \(A\) and \(B\) such that \(A B\) is not a subgroup of \(G\)

Let \(p\) be a prime number and \(G\) be a group with \(p^{a} k\) elements, where \(a\) is a positive integer and \(p\) does not divide \(k\). Suppose that \(P\) is a subgroup of \(G\) with \(p^{a}\) elements and \(Q\) is a subgroup of \(G\) with \(p^{b}\) elements, where \(0

Show that two right cosets \(H x, H y\) of a subgroup \(H\) in a group \(G\) are equal if and only if \(y x^{-1}\) is an element of \(H\).

Show that if an element \(y\) of a group \(G\) is in the right coset \(H x\) then \(H y=H x\)
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