Group Theory: Element of Order 2 in Groups of Even Order

In summary, the conversation discusses how to show that a group of even order has an element g not equal to the identity such that g^2=1. The solution involves pairing each element with its inverse and proving that if ab=1, then ba=1.
  • #1
PsychonautQQ
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Homework Statement


If G is a group of even order, show that it has an element g not equal to the identity such that g^2 = 1.

Homework Equations


None

The Attempt at a Solution


What I wrote:

If |G| = n, then g^n = 1 for some g in G. Thus, (g^(n/2))(g^(n/2)) = 1, so g^(n/2) is the element of order 2.

Is this a flawed argument? there guaranteed to be an element such that g^n = 1?
 
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  • #2
The identity is, of course, its own inverse. Since the group is "of even order", i.e. it has an even number of elements, removing the identity leaves an odd number of elements. Pairing each number with its inverse, what happens?
 
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  • #3
HallsofIvy said:
The identity is, of course, its own inverse. Since the group is "of even order", i.e. it has an even number of elements, removing the identity leaves an odd number of elements. Pairing each number with its inverse, what happens?
I see your arguement. Is this true though? Can't ab = 1, bc = 1, cd = 1 etc etc but ba not equal 1?
 
  • #4
PsychonautQQ said:
I see your arguement. Is this true though? Can't ab = 1, bc = 1, cd = 1 etc etc but ba not equal 1?

No, it's not possible. If ab=1 then ba=1. Prove it!
 
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1. What is Group Theory?

Group Theory is a mathematical discipline that studies the properties and structures of groups, which are sets of elements that follow certain operations and axioms.

2. What is a group?

A group is a set of elements that is closed under a binary operation, associative, has an identity element, and each element has an inverse.

3. What are the applications of Group Theory?

Group Theory has applications in various fields such as physics, chemistry, cryptography, computer science, and engineering. It is used to study symmetries, patterns, and structures in different systems and phenomena.

4. What are some basic concepts in Group Theory?

Some basic concepts in Group Theory include group operations, subgroups, cosets, normal subgroups, homomorphisms, and isomorphisms.

5. How is Group Theory related to symmetry?

Group Theory is closely related to symmetry because symmetries can be described and studied using group theory concepts. Groups can represent the symmetries of an object or system, and group operations can be used to analyze and classify symmetries.

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