A little problem with permutations.

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SUMMARY

The discussion focuses on the subgroup A_6 of S_6, which consists of all even permutations. It addresses the question of how many permutations of order 6 are included in A_6. The order of a permutation is defined as the least common multiple (lcm) of the lengths of its disjoint cycles. It is concluded that any element of S_6 with order 6 must be odd, indicating that A_6 contains no permutations of order 6.

PREREQUISITES
  • Understanding of group theory and permutation groups
  • Familiarity with even and odd permutations
  • Knowledge of disjoint cycle decomposition
  • Concept of least common multiple (lcm)
NEXT STEPS
  • Study the properties of symmetric groups, specifically S_n and A_n
  • Learn about cycle notation and its applications in group theory
  • Explore the implications of permutation orders in algebraic structures
  • Investigate the relationship between even and odd permutations in various groups
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Mathematicians, students of abstract algebra, and anyone interested in the properties of permutation groups and their applications in group theory.

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Let A_n be a subgroup of S_n that includes all the even permutations. How many permutations of order 6 does A_6 include?
 
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What do you mean by "order of a permutation"?
 
The order of the permutation as an element of a group, of course.

Remember that if you decompose an element x of Sn as a product of disjoint cycles, then the order of x is equal to the lcm of the lengths of the cycles. You will find that if you decompose an element x of S6 of order 6 into disjoint cycles, then x must be odd.
 

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