|G|=4. Prove the group is either cyclic or g^2=e

mathmajor2013 · Feb 24, 2011

Let G be a group with |G|=4. Prove that either G is cyclic or for any x in G, x^2=e.

disregardthat · Feb 24, 2011

Consider and element x in G of maximum order. What possibilities are there, and what does each of them say about the group structure?

mathmajor2013 · Feb 24, 2011

So the order of x is 2 or 4, right? Since the order must divide 4. If it's 2, the x^2=e, and so each element is its own inverse. If it's 4...I'm lost.

disregardthat · Feb 24, 2011

Correct. And if it's 4, what can you say about x, x^2,x^3 and x^4? Are any of them equal? What is x for this group?

mathmajor2013 · Feb 24, 2011

Damn. They are all unique, and x^4=e. and x is the generator for the group yeah?

mathmajor2013 · Feb 24, 2011

So the order of x is 2 or 4, right? Since the order must divide 4. If it's 2, the x^2=e, and so each element is its own inverse. If it's 4...I'm lost.

Robert1986 · Feb 25, 2011

You are correct. The order of each element in the group must divide the order of the group. So, the order of each element must be 1,2 or 4. If there are no elements of order 4, then x^2=e for each x in the group (since the order of each element is 1 or 2). If there is an element of order 4, then this element is a generator, as you pointed out, and so the group is cyclic.

lavinia · Feb 25, 2011

mathmajor2013 said:

Let G be a group with |G|=4. Prove that either G is cyclic or for any x in G, x^2=e.

a non identity element must have either order 4 or order 2

|G|=4. Prove the group is either cyclic or g^2=e

Discussion

Undergrad Spinor calculus

Undergrad Matrix representation of rank-2 spinors

Undergrad Looking for a paper about spinors

Undergrad Why are these cycles disjoint and of equal length?

On the Moore–Penrose inverse from a banal linear algebra viewpoint

Undergrad Correct constructed example of Abelian Monoid?

Undergrad Correct state row entries in the following Sequential machine?

Insights Why Vector Spaces Explain The World: A Historical Perspective

Undergrad Quintic Polynomial Tschirnhaus Transform: Possible complex Bring Radicals

Insights Revisiting the Velocity-Time Function

Insights Remote Operated Gate Control System

Insights AI Enriched Problem Solving

Insights Thinking Outside The Box Versus Knowing What’s In The Box

Insights Why Entangled Photon-Polarization Qubits Violate Bell’s Inequality

Insights Quantum Entanglement is a Kinematic Fact, not a Dynamical Effect

|G|=4. Prove the group is either cyclic or g^2=e

Discussion

Similar threads

Undergrad How do I prove a group is cyclic without Sylow's theorems?

Graduate How to Prove a Group with 2n-1 Subgroups is Trivial or Isomorphic to Z2?

Graduate If p is prime, prove the group (Zp*,x) has exactly one element of order 2.

Graduate Prove that (a1*a2*...*an)^2 = e in a finite abelian group

Prove G is Cyclic: Prime p Order Group

Graduate Elements of Subgroups in Additive Group Q/Z: G(2) and G(P)

How to Prove a Group is Cyclic?

Proving a Group is Cyclic: What is the Generator of G?

G is cyclic and |G| = p^n, p is prime <=> H,K Subgroups, H⊆K or K⊆H

Graduate How can we use induction to prove that (xy)^{3} = y^{3}x^{3} for all x,y in G?

Graduate Prove that (a1a2...*an)^2 = e in a finite abelian group