What Are the Subgroups and Normal Subgroups of D4?

  • Thread starter Thread starter Trigger
  • Start date Start date
  • Tags Tags
    Algebra
Click For Summary
SUMMARY

The discussion focuses on identifying the subgroups and normal subgroups of the dihedral group D4, which consists of the elements {e, t, t², t³, s, st, st², st³}. It establishes that any subgroup containing half the elements of D4 is a normal subgroup, and highlights that the subgroup {e, R²} is indeed a normal subgroup. The conversation emphasizes the importance of understanding group notation and properties, particularly regarding the order of elements and their implications for subgroup structure.

PREREQUISITES
  • Understanding of group theory concepts, specifically dihedral groups.
  • Familiarity with subgroup properties and normal subgroups.
  • Knowledge of Cayley tables and their construction.
  • Basic understanding of element orders within groups.
NEXT STEPS
  • Study the properties of dihedral groups, particularly D4 and its structure.
  • Learn how to construct and interpret Cayley tables for various groups.
  • Explore the criteria for identifying normal subgroups in finite groups.
  • Investigate the implications of element orders on subgroup formation.
USEFUL FOR

Students and researchers in abstract algebra, particularly those focusing on group theory, as well as educators teaching the concepts of dihedral groups and subgroup structures.

Trigger
Messages
11
Reaction score
0
Calculate all the subgroups of D_4.
Which of them are normal subgroups? (It can
be shown that any subgroup containing half
the elements of a group G is a normal
subgroup, and if a has order 2 then {e,a} is
a normal subgroup iff a commutes with all
elements of G.)

{e,R^2} happens to be a normal subgroup.
Give the Cayley table of D_4/{e,R^2}.
 
Physics news on Phys.org
I have made out a Cayley Table for D4, but do not know how to calculate the subgroups. Please help!
 
What the heck is R?

Let D_4 be e,t,t^2,t^3,s,st,st^2,st^3 t the rotation s the reflection (see how explaining notation can help?)

Let H be a proper subgroup. if H contains t, then it contains all its powers, and has order 4. It can contain no other elements as its order must divide 8, and hence would contain all elements.

If H doesn't contain t then... do some thinking and use the definitions of things.
 

Similar threads

Undergrad Nontrivial Normal Subgroup in Finite Groups with Index Constraints?
  • · Replies 1 ·
Replies
1
Views
3K
Undergrad Is G simple?Is G simple? A different approach using Sylow theorems
  • · Replies 5 ·
Replies
5
Views
3K
Graduate Near-Rings with Noncommutative Addition and Two-Sided Distributivity
  • · Replies 4 ·
Replies
4
Views
3K
Normal subgroup generated by a subset A
  • · Replies 15 ·
Replies
15
Views
3K
Undergrad Subgroup axioms for a symmetric group
  • · Replies 1 ·
Replies
1
Views
1K
Showing that subgroup of unique order implies normality
  • · Replies 3 ·
Replies
3
Views
2K
Having all subgroups normal is isomorphism invariant
  • · Replies 1 ·
Replies
1
Views
2K
Subgroup of Index ##n## for every ##n \in \Bbb{N}##.
  • · Replies 1 ·
Replies
1
Views
2K
MHB Normal series and composition series
  • · Replies 8 ·
Replies
8
Views
2K
Normal group of order 60 isomorphic to A_5
  • · Replies 6 ·
Replies
6
Views
2K