Subgroup axioms for a symmetric group

In summary, the textbook is asking for a proof using subgroup axioms to determine whether a set of permutations that interchange two specific symbols in S4 is a subgroup of the symmetric group, as well as for a set of permutations that fix two elements. It is determined that the set that interchanges symbols is not a subgroup due to a lack of the identity element, while the set that fixes symbols does contain the identity element and satisfies axiom 2. It is suggested to either formally prove each axiom or provide a counterexample to demonstrate why the axioms are or are not satisfied. It is advised to write down all permutations that meet the criteria in order to fully understand and prove the subgroup status.
  • #1
penroseandpaper
21
0
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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  • #2
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.
 

1. What is a symmetric group?

A symmetric group is a mathematical group that consists of all possible permutations of a finite set of elements. In other words, it is a group of symmetries that can be applied to a set of objects.

2. What are subgroup axioms?

Subgroup axioms are a set of rules or properties that a subset of a group must satisfy in order to be considered a subgroup. These axioms ensure that the subset behaves like a group on its own.

3. How many subgroup axioms are there for a symmetric group?

There are three main subgroup axioms for a symmetric group: closure, associativity, and identity. These axioms state that the result of combining any two elements in the subgroup must also be in the subgroup, the order in which elements are combined does not matter, and there must be an element in the subgroup that acts as an identity element.

4. Why are subgroup axioms important for a symmetric group?

Subgroup axioms are important because they allow us to study smaller, more manageable groups within a larger group. They also help us understand the structure and properties of a group by breaking it down into smaller subsets.

5. Are there any other important properties of a subgroup in a symmetric group?

Yes, there are two additional properties that a subgroup in a symmetric group must satisfy: inverses and closure under inverses. These properties state that for every element in the subgroup, there must be an inverse element also in the subgroup, and the result of combining an element with its inverse must also be in the subgroup.

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