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Let G be a group of odd order. Show that for all a E G there exists b E G such that a = b?.

Let G be a group of odd order. Show that for all a E G there exists b E G such that a = b?.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Could you please explain how to show this in detail? I don't understand if it is too concise... 

**Problem 7.** Let \( G \) be a group of odd order. Show that for all \( a \in G \) there exists \( b \in G \) such that \( a = b^2 \). *Explanation:* This problem explores a property of groups in abstract algebra, specifically involving elements and their squares in a group of odd order. The goal is to demonstrate that every element \( a \) in the group \( G \) can be expressed as the square of some element \( b \) from the same group. Approach this by considering the properties of group elements, potential symmetry, and using the fact that the order of the group is odd.
Transcribed Image Text:**Problem 7.** Let \( G \) be a group of odd order. Show that for all \( a \in G \) there exists \( b \in G \) such that \( a = b^2 \). *Explanation:* This problem explores a property of groups in abstract algebra, specifically involving elements and their squares in a group of odd order. The goal is to demonstrate that every element \( a \) in the group \( G \) can be expressed as the square of some element \( b \) from the same group. Approach this by considering the properties of group elements, potential symmetry, and using the fact that the order of the group is odd.
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