Counter Example: Abelian Subgroup Not Normal

  • Thread starter Thread starter Bachelier
  • Start date Start date
  • Tags Tags
    Normal
Click For Summary
A counterexample of an abelian subgroup that is not normal can be found in the dihedral group D_6, which represents the symmetries of a triangle. The subgroup generated by a single rotation, such as {e, r}, where r is a rotation, is abelian but not normal in D_6. This is because the conjugate of a rotation by a reflection does not remain within the subgroup. Thus, while the subgroup is abelian, it fails to meet the criteria for normality. This illustrates that not all abelian subgroups are normal within a group.
Bachelier
Messages
375
Reaction score
0
--------------------------------------------------------------------------------

Can someone provide us with a counter example to an abelian (sub)group that is not normal.

I'm thinking something in the center of a group.
 
Physics news on Phys.org
Can you find a subgroup in D_6?? (the dihedral group of order 6, aka the symmetry group of a triangle). Can you find a subgroup that is not normal?? Is this subgroup abelian?
 

Thread 'Confusion about the Moyal-Weyl twist'

I am studying the mathematical formalism behind non-commutative geometry approach to quantum gravity. I was reading about Hopf algebras and their Drinfeld twist with a specific example of the Moyal-Weyl twist defined as F=exp(-iλ/2θ^(μν)∂_μ⊗∂_ν) where λ is a constant parametar and θ antisymmetric constant tensor. {∂_μ} is the basis of the tangent vector space over the underlying spacetime Now, from my understanding the enveloping algebra which appears in the definition of the Hopf algebra...
View full post »

Similar threads

Graduate Near-Rings with Noncommutative Addition and Two-Sided Distributivity
  • · Replies 4 ·
Replies
4
Views
2K
Undergrad Differences between left vs right actions in some group theory questions
  • · Replies 1 ·
Replies
1
Views
523
MHB Z6: Identity, Order, Inverse, Generator, Abelian/Non-Abelian, Subgroups
  • · Replies 4 ·
Replies
4
Views
3K
How Is the Abelianization of a Lie Group Defined?
  • · Replies 3 ·
Replies
3
Views
2K
MHB G contains a normal p-Sylow subgroup
  • · Replies 5 ·
Replies
5
Views
2K
A few questions on abelian and normal subgroups
  • · Replies 4 ·
Replies
4
Views
3K
MHB Can You Prove These Properties of Normal Subgroups in Group Theory?
  • · Replies 2 ·
Replies
2
Views
2K
Undergrad Is G simple?Is G simple? A different approach using Sylow theorems
  • · Replies 5 ·
Replies
5
Views
2K
Is G/H Always an Abelian Group if H is Normal in G?
  • · Replies 1 ·
Replies
1
Views
2K
Undergrad Decomposition per the Fundamental Theorem of Finite Abelian Groups
  • · Replies 13 ·
Replies
13
Views
3K