A group is an algebraic structure that is closed under binary operation, associative, each of its element has an inverse and there is a common identity element.
Problem
Let be a finite group of even order. Show that has an element such tha
Proof
Let be the group of even order.
Suppose that each element of is paired with its inverse ( Note that the identity element is paired to itself)
Then consists of the union of all of the pairs
Now,
Suppose that no element other than identity, is paired to itself
Let the number of pairs excluding the identity be , then (since there are pairs + the identity)
But, this makes the order of odd, a contradiction, and thus there must be at least one element that is paired with its inverse (this makes the order of even)
Let be paired with its inverse, then , and hence
-
- Active Topics
-
-
- by Eli 9 hours ago Pondering Big Cosmology Questions Through Lectures and Dialogues View the latest post Replies 34 Views 55369
- by Eli 9 hours ago Russia Invades Ukraine View the latest post Replies 652 Views 220555
- by Eli 9 hours ago Re: What is in Your Mind? View the latest post Replies 693 Views 284754
- by Eli 1 day ago Programmatically Manipulate Files: Renaming, Reading, Writing, Deleting, and Moving Files Between Folders View the latest post Replies 7 Views 2916
- by Eli 6 days ago Iran Launches Retaliatory Attack Against Israel, and Israel Retaliates by Attacking Iranian Isfahan Millitary Base View the latest post Replies 28 Views 2506
- by Eli 1 week ago All in One: YouTube, TED, X, Facebook and Instagram Reels, Videos, Images and Text Posts View the latest post Replies 319 Views 19644
- by Eli 1 week ago Python Packages for Scientific Computing View the latest post Replies 8 Views 11237
- by Eli 1 week ago Dunia Yetu: Building Tanzania's Digital Future Together View the latest post Replies 5 Views 2002
- by Eli 2 weeks ago Collection of Greatest Christian Hymns of all Times View the latest post Replies 33 Views 53798
- by Eli 3 weeks ago What is Retrieval-Augmented Generation (RAG)? View the latest post Replies 2 Views 1691
-
A Group of Even Order
-
- Information
-
Who is online
Users browsing this forum: No registered users and 0 guests