Prove Subset can't be coset with two different subgroups

  • Thread starter Thread starter FanofAFan
  • Start date Start date
Click For Summary
SUMMARY

The discussion centers on proving that a subset S of a group G cannot be a right coset of two different subgroups of G. The user presents an initial approach involving fixing an element z in the subgroup H and exploring the implications of the right coset equation. The conclusion drawn is that if the intersection of two right cosets is non-empty, then the two cosets must be equal, thereby reinforcing the claim that a subset cannot simultaneously represent two distinct right cosets.

PREREQUISITES
  • Understanding of group theory concepts, specifically right cosets.
  • Familiarity with subgroup properties in abstract algebra.
  • Knowledge of intersection properties of sets.
  • Basic proficiency in mathematical proof techniques.
NEXT STEPS
  • Study the definition and properties of right cosets in group theory.
  • Explore the implications of the intersection of subgroups in abstract algebra.
  • Learn about the Lagrange's theorem and its applications in group theory.
  • Review examples of groups and their subgroups to solidify understanding of cosets.
USEFUL FOR

Mathematics students, particularly those studying abstract algebra, and educators looking to deepen their understanding of group theory concepts related to cosets and subgroups.

FanofAFan
Messages
43
Reaction score
0
Prove that a subset S of a group G cannot be a right coset of two different subgroups of G.

I having a hard time proving this... but this is what I have so far;
Fix z in Hz and since z = ez where e is in H. Then assume z in Hy by the right coset equation then with x = z we see that Hz = Hy and if Hx intersection of Hy [tex]\neq[/tex] to the empty set... that as far as i got. Please help
 
Physics news on Phys.org

Similar threads

Is H a Normal Subgroup If Every Left Coset Equals a Right Coset?
  • · Replies 1 ·
Replies
1
Views
3K
Finding Cosets of subgroup <(3,2,1)> of G = S3
  • · Replies 4 ·
Replies
4
Views
4K
Write down the cosets (right/left) of this:
  • · Replies 21 ·
Replies
21
Views
3K
Show group equivalence relation associated with normal subgroup
  • · Replies 6 ·
Replies
6
Views
3K
Subgroup of Index ##n## for every ##n \in \Bbb{N}##.
  • · Replies 1 ·
Replies
1
Views
2K
Proving something about right cosets of distinct subgroups of a group
  • · Replies 1 ·
Replies
1
Views
4K
Question about Cosets: Does abH=baH imply ab=ba? | Group Theory Homework
  • · Replies 2 ·
Replies
2
Views
2K
Undergrad Differences between left vs right actions in some group theory questions
  • · Replies 1 ·
Replies
1
Views
1K
Prove that no group of order 160 is simple
  • · Replies 2 ·
Replies
2
Views
4K
Normal Subgroups and Isomorphisms
  • · Replies 9 ·
Replies
9
Views
2K