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Let S3 be the group of all permutation with 3 symbols then the number of elements in S3 that satisfy the equation x2 = e (where e is identity) is (Answer should be integer) __________.
Correct answer is '4'. Can you explain this answer?
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Let S3be the group of all permutation with 3 symbols then the number o...
Understanding S3 and Permutations
S3, the symmetric group of degree 3, consists of all possible permutations of three symbols (for example, {1, 2, 3}). The group has a total of 3! (factorial of 3) = 6 elements.
Elements of S3
The elements in S3 can be listed as follows:
- (1) : Identity (e)
- (12) : Swap 1 and 2
- (13) : Swap 1 and 3
- (23) : Swap 2 and 3
- (123) : Cycle 1 to 2, 2 to 3, and 3 to 1
- (132) : Cycle 1 to 3, 3 to 2, and 2 to 1
Condition x² = e
To find the elements satisfying x² = e, we need permutations where applying the permutation twice results in the identity permutation.
Identifying the Elements
1. Identity (1): Applying it twice gives (1), satisfying x² = e.
2. Transpositions (12), (13), (23): Each of these swaps two elements. When applied twice, they revert back to the identity:
- (12)(12) = (1)
- (13)(13) = (1)
- (23)(23) = (1)
3. 3-Cycles (123) and (132): When applied twice, they do not return to the identity:
- (123)(123) = (132) ≠ (1)
- (132)(132) = (123) ≠ (1)
Conclusion
The elements satisfying x² = e in S3 are:
- Identity (1)
- Transposition (12)
- Transposition (13)
- Transposition (23)
Thus, the total number of elements in S3 that satisfy the equation x² = e is 4.
S3, the symmetric group of degree 3, consists of all possible permutations of three symbols (for example, {1, 2, 3}). The group has a total of 3! (factorial of 3) = 6 elements.
Elements of S3
The elements in S3 can be listed as follows:
- (1) : Identity (e)
- (12) : Swap 1 and 2
- (13) : Swap 1 and 3
- (23) : Swap 2 and 3
- (123) : Cycle 1 to 2, 2 to 3, and 3 to 1
- (132) : Cycle 1 to 3, 3 to 2, and 2 to 1
Condition x² = e
To find the elements satisfying x² = e, we need permutations where applying the permutation twice results in the identity permutation.
Identifying the Elements
1. Identity (1): Applying it twice gives (1), satisfying x² = e.
2. Transpositions (12), (13), (23): Each of these swaps two elements. When applied twice, they revert back to the identity:
- (12)(12) = (1)
- (13)(13) = (1)
- (23)(23) = (1)
3. 3-Cycles (123) and (132): When applied twice, they do not return to the identity:
- (123)(123) = (132) ≠ (1)
- (132)(132) = (123) ≠ (1)
Conclusion
The elements satisfying x² = e in S3 are:
- Identity (1)
- Transposition (12)
- Transposition (13)
- Transposition (23)
Thus, the total number of elements in S3 that satisfy the equation x² = e is 4.
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Community Answer
Let S3be the group of all permutation with 3 symbols then the number o...
If x2 = e then either o(x) is lor 2.
We know that the number of elements of order 2 in S3 = 3.
Thus the mumber ofelements in S3 that satisfy the equation.x2 = e is 4.
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Let S3be the group of all permutation with 3 symbols then the number of elements in S3that satisfy theequation x2= e (where e is identity) is (Answer should be integer) __________.Correct answer is '4'. Can you explain this answer?
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