Subgroup axioms for a symmetric group

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SUMMARY

The discussion centers on the application of subgroup axioms to determine whether specific sets of permutations in S4 form subgroups of the symmetric group. The set of permutations that interchange two symbols does not qualify as a subgroup because it lacks the identity element, while the set that fixes two symbols does qualify as it contains the identity element. Participants emphasize the importance of verifying all four group axioms—closure, identity, inverses, and associativity—when proving subgroup status. Additionally, calculations reveal four permutations for the interchanging set and two for the fixing set, prompting further inquiry into completeness.

PREREQUISITES
  • Understanding of group theory concepts, specifically subgroup axioms.
  • Familiarity with symmetric groups, particularly S4.
  • Knowledge of permutation notation and cycle notation.
  • Ability to perform basic group operations and verify group properties.
NEXT STEPS
  • Formally prove each of the four subgroup axioms for the identified sets of permutations.
  • Explore the concept of identity elements in symmetric groups.
  • Investigate counterexamples for permutations that do not satisfy subgroup criteria.
  • Study the implications of closure and inverses in the context of symmetric groups.
USEFUL FOR

Students of abstract algebra, particularly those studying group theory, as well as educators and anyone seeking to deepen their understanding of symmetric groups and subgroup properties.

penroseandpaper
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Hi,

The textbook asks me to use subgroup axioms to prove why a set of permutations that interchange two specific symbols in S4 is or isn't a subgroup of the symmetric group, and the same for a set of permutations that fix two elements.

My guess is that the set of permutations that interchange the two symbols isn't a subgroup under such rules because it doesn't contain the identity element. Meanwhile, the one that fixes the two symbols does contain the identity element and hence satisfies axiom 2.

I was wondering if I'm right in saying that, and whether I need to consider either of the two other axioms in proving it. My thoughts are it's otherwise associative and inverses contained.

My calculations also produced four permutations in cycle form for the interchanging set and two permutations for fixing (identity and one more) - did I find them all?

Group theory as a lockdown challenge is proving a little trickier than expected! But it doesn't help that so many textbooks don't have any answers in them...
Still, it's nice to stretch the grey matter.

Sorry to bother you and thanks for your help,
Penn
 
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penroseandpaper said:
Hi,

The textbook asks me to use subgroup axioms to prove why a set of permutations that interchange two specific symbols in S4 is or isn't a subgroup of the symmetric group, and the same for a set of permutations that fix two elements.

My guess is that the set of permutations that interchange the two symbols isn't a subgroup under such rules because it doesn't contain the identity element. Meanwhile, the one that fixes the two symbols does contain the identity element and hence satisfies axiom 2.

I was wondering if I'm right in saying that, and whether I need to consider either of the two other axioms in proving it. My thoughts are it's otherwise associative and inverses contained.

My calculations also produced four permutations in cycle form for the interchanging set and two permutations for fixing (identity and one more) - did I find them all?

Group theory as a lockdown challenge is proving a little trickier than expected! But it doesn't help that so many textbooks don't have any answers in them...
Still, it's nice to stretch the grey matter.

Sorry to bother you and thanks for your help,
Penn
You should either: a) formally prove each axiom holds; or, b) provide a concrete counterexample of which axioms fail.

It's a good exercise to cover all four group axioms and show why each is or is not satisfied.

In each case, it would help to write down all permutations that meet the criteria.
 

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