Unique Elements from Subsets of Symmetric Group
$begingroup$
Good morning to you all! I have math question, plz help me out!
We take a subset $A$ from a symmetric group $S_n$ (n elements).
Now consider all elements of $B$ composed of $a * b$ where $*$ is the group operation of symmetric group and $a, b in A$ ($a, b$ are elements of $A$).
$$B={e| e=a*b, text{where} quad a,b in A}$$
What sufficient conditions need to be imposed on set $A$ so $a * b neq c*d$ for any $a,b,c,d in A$ ?
PS:
$a=b$ is allowed so, $a*a in B$.
Note, we are composing $B$ using exactly two elements from $A$ under symmetric group operation.
Of course, we are not considering trivial cases like when the set has only one element, e.g. $a=b=c=d$.
you are welcome to give examples and counter examples.
abstract-algebra group-theory symmetric-groups
asked Jan 14 at 13:30
AndrewAndrew
63
$endgroup$
add a comment |
$begingroup$
Good morning to you all! I have math question, plz help me out!
We take a subset $A$ from a symmetric group $S_n$ (n elements).
Now consider all elements of $B$ composed of $a * b$ where $*$ is the group operation of symmetric group and $a, b in A$ ($a, b$ are elements of $A$).
$$B={e| e=a*b, text{where} quad a,b in A}$$
What sufficient conditions need to be imposed on set $A$ so $a * b neq c*d$ for any $a,b,c,d in A$ ?
PS:
$a=b$ is allowed so, $a*a in B$.
Note, we are composing $B$ using exactly two elements from $A$ under symmetric group operation.
Of course, we are not considering trivial cases like when the set has only one element, e.g. $a=b=c=d$.
you are welcome to give examples and counter examples.
abstract-algebra group-theory symmetric-groups
asked Jan 14 at 13:30
AndrewAndrew
63
$endgroup$
2
$begingroup$
"What sufficient conditions need to be imposed on set A so $a∗b neq c∗d$ for any $a,b,c,d in A$ ?" Are you sure about this wording? If I choose $a=b=c=d$ then obviously there is equality, which leads to the trivial solution.
$endgroup$
– Mariuslp
Jan 14 at 13:37
3
$begingroup$
Your edit (adding 3) does not address @Mariuslp 's question. Until you clarify what you mean this is unanswerable. Also in line 1, what has $n$ elements?
$endgroup$
– ancientmathematician
Jan 14 at 16:12
add a comment |
$begingroup$
Good morning to you all! I have math question, plz help me out!
We take a subset $A$ from a symmetric group $S_n$ (n elements).
Now consider all elements of $B$ composed of $a * b$ where $*$ is the group operation of symmetric group and $a, b in A$ ($a, b$ are elements of $A$).
$$B={e| e=a*b, text{where} quad a,b in A}$$
What sufficient conditions need to be imposed on set $A$ so $a * b neq c*d$ for any $a,b,c,d in A$ ?
PS:
$a=b$ is allowed so, $a*a in B$.
Note, we are composing $B$ using exactly two elements from $A$ under symmetric group operation.
Of course, we are not considering trivial cases like when the set has only one element, e.g. $a=b=c=d$.
you are welcome to give examples and counter examples.
abstract-algebra group-theory symmetric-groups
asked Jan 14 at 13:30
AndrewAndrew
63
$endgroup$
Good morning to you all! I have math question, plz help me out!
We take a subset $A$ from a symmetric group $S_n$ (n elements).
Now consider all elements of $B$ composed of $a * b$ where $*$ is the group operation of symmetric group and $a, b in A$ ($a, b$ are elements of $A$).
$$B={e| e=a*b, text{where} quad a,b in A}$$
What sufficient conditions need to be imposed on set $A$ so $a * b neq c*d$ for any $a,b,c,d in A$ ?
PS:
$a=b$ is allowed so, $a*a in B$.
Note, we are composing $B$ using exactly two elements from $A$ under symmetric group operation.
Of course, we are not considering trivial cases like when the set has only one element, e.g. $a=b=c=d$.
you are welcome to give examples and counter examples.
abstract-algebra group-theory symmetric-groups
abstract-algebra group-theory symmetric-groups
asked Jan 14 at 13:30
AndrewAndrew
63
asked Jan 14 at 13:30
AndrewAndrew
63
edited Jan 15 at 1:49
Andrew
asked Jan 14 at 13:30
AndrewAndrew
63
asked Jan 14 at 13:30
AndrewAndrew
63
asked Jan 14 at 13:30
AndrewAndrew
63
63
2
$begingroup$
"What sufficient conditions need to be imposed on set A so $a∗b neq c∗d$ for any $a,b,c,d in A$ ?" Are you sure about this wording? If I choose $a=b=c=d$ then obviously there is equality, which leads to the trivial solution.
$endgroup$
– Mariuslp
Jan 14 at 13:37
3
$begingroup$
Your edit (adding 3) does not address @Mariuslp 's question. Until you clarify what you mean this is unanswerable. Also in line 1, what has $n$ elements?
$endgroup$
– ancientmathematician
Jan 14 at 16:12
add a comment |
2
$begingroup$
"What sufficient conditions need to be imposed on set A so $a∗b neq c∗d$ for any $a,b,c,d in A$ ?" Are you sure about this wording? If I choose $a=b=c=d$ then obviously there is equality, which leads to the trivial solution.
$endgroup$
– Mariuslp
Jan 14 at 13:37
3
$begingroup$
Your edit (adding 3) does not address @Mariuslp 's question. Until you clarify what you mean this is unanswerable. Also in line 1, what has $n$ elements?
$endgroup$
– ancientmathematician
Jan 14 at 16:12
2
2
$begingroup$
"What sufficient conditions need to be imposed on set A so $a∗b neq c∗d$ for any $a,b,c,d in A$ ?" Are you sure about this wording? If I choose $a=b=c=d$ then obviously there is equality, which leads to the trivial solution.
$endgroup$
– Mariuslp
Jan 14 at 13:37
$begingroup$
"What sufficient conditions need to be imposed on set A so $a∗b neq c∗d$ for any $a,b,c,d in A$ ?" Are you sure about this wording? If I choose $a=b=c=d$ then obviously there is equality, which leads to the trivial solution.
$endgroup$
– Mariuslp
Jan 14 at 13:37
3
3
$begingroup$
Your edit (adding 3) does not address @Mariuslp 's question. Until you clarify what you mean this is unanswerable. Also in line 1, what has $n$ elements?
$endgroup$
– ancientmathematician
Jan 14 at 16:12
$begingroup$
Your edit (adding 3) does not address @Mariuslp 's question. Until you clarify what you mean this is unanswerable. Also in line 1, what has $n$ elements?
$endgroup$
– ancientmathematician
Jan 14 at 16:12
add a comment |
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$begingroup$
"What sufficient conditions need to be imposed on set A so $a∗b neq c∗d$ for any $a,b,c,d in A$ ?" Are you sure about this wording? If I choose $a=b=c=d$ then obviously there is equality, which leads to the trivial solution.
$endgroup$
– Mariuslp
Jan 14 at 13:37
3
$begingroup$
Your edit (adding 3) does not address @Mariuslp 's question. Until you clarify what you mean this is unanswerable. Also in line 1, what has $n$ elements?
$endgroup$
– ancientmathematician
Jan 14 at 16:12